# Polynomial Antiderivative and Constant Foundations

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261007-advance-20261008-r06-antiderivative-foundations · 2026-10-07 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Zero-Constant Antiderivative of a Monomial

On all R, f(x)=(0)x^0, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.

View answer and explanation[0/1]

Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0001/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-monomial · candidate





## 2. Zero-Constant Antiderivative of a Monomial

On all R, f(x)=(-2)x^1, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.

View answer and explanation[0/1, 0/1, -1/1]

Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, -1/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0002/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-monomial · candidate





## 3. Zero-Constant Antiderivative of a Monomial

On all R, f(x)=(3)x^2, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.

View answer and explanation[0/1, 0/1, 0/1, 1/1]

Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 1/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0003/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-monomial · candidate





## 4. Zero-Constant Antiderivative of a Monomial

On all R, f(x)=(-4)x^3, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.

View answer and explanation[0/1, 0/1, 0/1, 0/1, -1/1]

Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 0/1, -1/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0004/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-monomial · candidate





## 5. Zero-Constant Antiderivative of a Monomial

On all R, f(x)=(5)x^4, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.

View answer and explanation[0/1, 0/1, 0/1, 0/1, 0/1, 1/1]

Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 0/1, 0/1, 1/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0005/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-monomial · candidate





## 6. Zero-Constant Antiderivative of a Monomial

On all R, f(x)=(-6)x^5, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.

View answer and explanation[0/1, 0/1, 0/1, 0/1, 0/1, 0/1, -1/1]

Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 0/1, 0/1, 0/1, -1/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0006/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-monomial · candidate





## 7. Zero-Constant Antiderivative of a Monomial

On all R, f(x)=(7)x^0, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.

View answer and explanation[0/1, 7/1]

Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 7/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0007/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-monomial · candidate





## 8. Zero-Constant Antiderivative of a Monomial

On all R, f(x)=(-8)x^1, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.

View answer and explanation[0/1, 0/1, -4/1]

Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, -4/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0008/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-monomial · candidate





## 9. Zero-Constant Antiderivative of a Monomial

On all R, f(x)=(9)x^2, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.

View answer and explanation[0/1, 0/1, 0/1, 3/1]

Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 3/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0009/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-monomial · candidate





## 10. Zero-Constant Antiderivative of a Monomial

On all R, f(x)=(-10)x^3, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.

View answer and explanation[0/1, 0/1, 0/1, 0/1, -5/2]

Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 0/1, -5/2]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0010/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-monomial · candidate





## 11. Zero-Constant Antiderivative of a Monomial

On all R, f(x)=(11)x^4, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.

View answer and explanation[0/1, 0/1, 0/1, 0/1, 0/1, 11/5]

Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 0/1, 0/1, 11/5]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0011/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-monomial · candidate





## 12. Zero-Constant Antiderivative of a Monomial

On all R, f(x)=(-12)x^5, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.

View answer and explanation[0/1, 0/1, 0/1, 0/1, 0/1, 0/1, -2/1]

Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 0/1, 0/1, 0/1, -2/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0012/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-monomial · candidate





## 13. Zero-Constant Antiderivative of a Monomial

On all R, f(x)=(13)x^0, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.

View answer and explanation[0/1, 13/1]

Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 13/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0013/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-monomial · candidate





## 14. Zero-Constant Antiderivative of a Monomial

On all R, f(x)=(-14)x^1, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.

View answer and explanation[0/1, 0/1, -7/1]

Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, -7/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0014/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-monomial · candidate





## 15. Zero-Constant Antiderivative of a Monomial

On all R, f(x)=(15)x^2, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.

View answer and explanation[0/1, 0/1, 0/1, 5/1]

Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 5/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0015/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-monomial · candidate





## 16. Zero-Constant Antiderivative of a Monomial

On all R, f(x)=(-16)x^3, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.

View answer and explanation[0/1, 0/1, 0/1, 0/1, -4/1]

Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 0/1, -4/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0016/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-monomial · candidate





## 17. Zero-Constant Antiderivative of a Monomial

On all R, f(x)=(17)x^4, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.

View answer and explanation[0/1, 0/1, 0/1, 0/1, 0/1, 17/5]

Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 0/1, 0/1, 17/5]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0017/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-monomial · candidate





## 18. Zero-Constant Antiderivative of a Monomial

On all R, f(x)=(-18)x^5, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.

View answer and explanation[0/1, 0/1, 0/1, 0/1, 0/1, 0/1, -3/1]

Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 0/1, 0/1, 0/1, -3/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0018/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-monomial · candidate





## 19. Zero-Constant Antiderivative of a Monomial

On all R, f(x)=(19)x^0, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.

View answer and explanation[0/1, 19/1]

Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 19/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0019/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-monomial · candidate





## 20. Zero-Constant Antiderivative of a Monomial

On all R, f(x)=(-20)x^1, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.

View answer and explanation[0/1, 0/1, -10/1]

Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, -10/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0020/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-monomial · candidate





## 21. Zero-Constant Antiderivative of a Monomial

On all R, f(x)=(21)x^2, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.

View answer and explanation[0/1, 0/1, 0/1, 7/1]

Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 7/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0021/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-monomial · candidate





## 22. Zero-Constant Antiderivative of a Monomial

On all R, f(x)=(-22)x^3, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.

View answer and explanation[0/1, 0/1, 0/1, 0/1, -11/2]

Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 0/1, -11/2]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0022/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-monomial · candidate





## 23. Zero-Constant Antiderivative of a Monomial

On all R, f(x)=(23)x^4, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.

View answer and explanation[0/1, 0/1, 0/1, 0/1, 0/1, 23/5]

Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 0/1, 0/1, 23/5]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0023/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-monomial · candidate





## 24. Zero-Constant Antiderivative of a Monomial

On all R, f(x)=(-24)x^5, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.

View answer and explanation[0/1, 0/1, 0/1, 0/1, 0/1, 0/1, -4/1]

Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 0/1, 0/1, 0/1, 0/1, 0/1, -4/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0024/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-monomial · candidate





## 25. Zero-Constant Antiderivative of a Monomial

On all R, f(x)=(25)x^0, with 0≤n≤5 and x^0=1 even at zero. Give ascending reduced rational coefficients of its zero-constant primitive.

View answer and explanation[0/1, 25/1]

Integrate the monomial coefficient to c/(n+1) at power n+1 and set the constant to zero, giving [0/1, 25/1]. At c=0 normalize to [0/1]; nonzero cases may have length seven. This representative does not discard the full constant family.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0025/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-monomial · candidate





## 26. Zero-Constant Antiderivative of a Polynomial

On all R, f has ascending coefficients [0], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.

View answer and explanation[0/1]

Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1]. The actual antiderivative family also includes adding a constant.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0026/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-polynomial · candidate





## 27. Zero-Constant Antiderivative of a Polynomial

On all R, f has ascending coefficients [3], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.

View answer and explanation[0/1, 3/1]

Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, 3/1]. The actual antiderivative family also includes adding a constant.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0027/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-polynomial · candidate





## 28. Zero-Constant Antiderivative of a Polynomial

On all R, f has ascending coefficients [-10, 3, 1, 3], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.

View answer and explanation[0/1, -10/1, 3/2, 1/3, 3/4]

Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, -10/1, 3/2, 1/3, 3/4]. The actual antiderivative family also includes adding a constant.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0028/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-polynomial · candidate





## 29. Zero-Constant Antiderivative of a Polynomial

On all R, f has ascending coefficients [-9, 4, -1, 4], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.

View answer and explanation[0/1, -9/1, 2/1, -1/3, 1/1]

Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, -9/1, 2/1, -1/3, 1/1]. The actual antiderivative family also includes adding a constant.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0029/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-polynomial · candidate





## 30. Zero-Constant Antiderivative of a Polynomial

On all R, f has ascending coefficients [-8, 5, 0, 1], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.

View answer and explanation[0/1, -8/1, 5/2, 0/1, 1/4]

Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, -8/1, 5/2, 0/1, 1/4]. The actual antiderivative family also includes adding a constant.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0030/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-polynomial · candidate





## 31. Zero-Constant Antiderivative of a Polynomial

On all R, f has ascending coefficients [-7, 6, 1, 2], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.

View answer and explanation[0/1, -7/1, 3/1, 1/3, 1/2]

Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, -7/1, 3/1, 1/3, 1/2]. The actual antiderivative family also includes adding a constant.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0031/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-polynomial · candidate





## 32. Zero-Constant Antiderivative of a Polynomial

On all R, f has ascending coefficients [-6, 7, -1, 3], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.

View answer and explanation[0/1, -6/1, 7/2, -1/3, 3/4]

Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, -6/1, 7/2, -1/3, 3/4]. The actual antiderivative family also includes adding a constant.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0032/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-polynomial · candidate





## 33. Zero-Constant Antiderivative of a Polynomial

On all R, f has ascending coefficients [-5, 8, 0, 4], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.

View answer and explanation[0/1, -5/1, 4/1, 0/1, 1/1]

Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, -5/1, 4/1, 0/1, 1/1]. The actual antiderivative family also includes adding a constant.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0033/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-polynomial · candidate





## 34. Zero-Constant Antiderivative of a Polynomial

On all R, f has ascending coefficients [-4, 9, 1, 1], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.

View answer and explanation[0/1, -4/1, 9/2, 1/3, 1/4]

Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, -4/1, 9/2, 1/3, 1/4]. The actual antiderivative family also includes adding a constant.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0034/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-polynomial · candidate





## 35. Zero-Constant Antiderivative of a Polynomial

On all R, f has ascending coefficients [-3, 10, -1, 2], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.

View answer and explanation[0/1, -3/1, 5/1, -1/3, 1/2]

Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, -3/1, 5/1, -1/3, 1/2]. The actual antiderivative family also includes adding a constant.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0035/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-polynomial · candidate





## 36. Zero-Constant Antiderivative of a Polynomial

On all R, f has ascending coefficients [-2, 11, 0, 3], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.

View answer and explanation[0/1, -2/1, 11/2, 0/1, 3/4]

Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, -2/1, 11/2, 0/1, 3/4]. The actual antiderivative family also includes adding a constant.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0036/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-polynomial · candidate





## 37. Zero-Constant Antiderivative of a Polynomial

On all R, f has ascending coefficients [-1, 12, 1, 4], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.

View answer and explanation[0/1, -1/1, 6/1, 1/3, 1/1]

Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, -1/1, 6/1, 1/3, 1/1]. The actual antiderivative family also includes adding a constant.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0037/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-polynomial · candidate





## 38. Zero-Constant Antiderivative of a Polynomial

On all R, f has ascending coefficients [0, 13, -1, 1], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.

View answer and explanation[0/1, 0/1, 13/2, -1/3, 1/4]

Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, 0/1, 13/2, -1/3, 1/4]. The actual antiderivative family also includes adding a constant.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0038/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-polynomial · candidate





## 39. Zero-Constant Antiderivative of a Polynomial

On all R, f has ascending coefficients [1, 14, 0, 2], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.

View answer and explanation[0/1, 1/1, 7/1, 0/1, 1/2]

Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, 1/1, 7/1, 0/1, 1/2]. The actual antiderivative family also includes adding a constant.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0039/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-polynomial · candidate





## 40. Zero-Constant Antiderivative of a Polynomial

On all R, f has ascending coefficients [2, 15, 1, 3], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.

View answer and explanation[0/1, 2/1, 15/2, 1/3, 3/4]

Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, 2/1, 15/2, 1/3, 3/4]. The actual antiderivative family also includes adding a constant.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0040/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-polynomial · candidate





## 41. Zero-Constant Antiderivative of a Polynomial

On all R, f has ascending coefficients [3, 16, -1, 4], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.

View answer and explanation[0/1, 3/1, 8/1, -1/3, 1/1]

Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, 3/1, 8/1, -1/3, 1/1]. The actual antiderivative family also includes adding a constant.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0041/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-polynomial · candidate





## 42. Zero-Constant Antiderivative of a Polynomial

On all R, f has ascending coefficients [4, 17, 0, 1], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.

View answer and explanation[0/1, 4/1, 17/2, 0/1, 1/4]

Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, 4/1, 17/2, 0/1, 1/4]. The actual antiderivative family also includes adding a constant.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0042/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-polynomial · candidate





## 43. Zero-Constant Antiderivative of a Polynomial

On all R, f has ascending coefficients [5, 18, 1, 2], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.

View answer and explanation[0/1, 5/1, 9/1, 1/3, 1/2]

Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, 5/1, 9/1, 1/3, 1/2]. The actual antiderivative family also includes adding a constant.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0043/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-polynomial · candidate





## 44. Zero-Constant Antiderivative of a Polynomial

On all R, f has ascending coefficients [6, 19, -1, 3], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.

View answer and explanation[0/1, 6/1, 19/2, -1/3, 3/4]

Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, 6/1, 19/2, -1/3, 3/4]. The actual antiderivative family also includes adding a constant.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0044/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-polynomial · candidate





## 45. Zero-Constant Antiderivative of a Polynomial

On all R, f has ascending coefficients [7, 20, 0, 4], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.

View answer and explanation[0/1, 7/1, 10/1, 0/1, 1/1]

Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, 7/1, 10/1, 0/1, 1/1]. The actual antiderivative family also includes adding a constant.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0045/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-polynomial · candidate





## 46. Zero-Constant Antiderivative of a Polynomial

On all R, f has ascending coefficients [8, 21, 1, 1], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.

View answer and explanation[0/1, 8/1, 21/2, 1/3, 1/4]

Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, 8/1, 21/2, 1/3, 1/4]. The actual antiderivative family also includes adding a constant.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0046/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-polynomial · candidate





## 47. Zero-Constant Antiderivative of a Polynomial

On all R, f has ascending coefficients [9, 22, -1, 2], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.

View answer and explanation[0/1, 9/1, 11/1, -1/3, 1/2]

Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, 9/1, 11/1, -1/3, 1/2]. The actual antiderivative family also includes adding a constant.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0047/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-polynomial · candidate





## 48. Zero-Constant Antiderivative of a Polynomial

On all R, f has ascending coefficients [10, 23, 0, 3], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.

View answer and explanation[0/1, 10/1, 23/2, 0/1, 3/4]

Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, 10/1, 23/2, 0/1, 3/4]. The actual antiderivative family also includes adding a constant.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0048/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-polynomial · candidate





## 49. Zero-Constant Antiderivative of a Polynomial

On all R, f has ascending coefficients [11, 24, 1, 4], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.

View answer and explanation[0/1, 11/1, 12/1, 1/3, 1/1]

Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, 11/1, 12/1, 1/3, 1/1]. The actual antiderivative family also includes adding a constant.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0049/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-polynomial · candidate





## 50. Zero-Constant Antiderivative of a Polynomial

On all R, f has ascending coefficients [12, 25, -1, 1], starting with its constant term. Give ascending reduced positive-denominator rational coefficients of the zero-constant primitive, removing trailing zero terms.

View answer and explanation[0/1, 12/1, 25/2, -1/3, 1/4]

Place a_k/(k+1) at power k+1, with constant zero, obtaining [0/1, 12/1, 25/2, -1/3, 1/4]. The actual antiderivative family also includes adding a constant.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0050/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-polynomial · candidate





## 51. Coefficient Eligibility of a Candidate Antiderivative

On all R, f has ascending coefficients [0], starting with its constant term. The candidate F has ascending rational coefficient pairs [[7, 1]], with positive denominators. Is F′=f as a coefficient identity?

View answer and explanationYes

Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving Yes. This is coefficient identity, not finite point sampling; the candidate constant is free.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0051/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-candidate · candidate





## 52. Coefficient Eligibility of a Candidate Antiderivative

On all R, f has ascending coefficients [-11, 2, 0, 2], starting with its constant term. The candidate F has ascending rational coefficient pairs [[-12, 1], [-11, 1], [1, 1], [0, 1], [3, 2]], with positive denominators. Is F′=f as a coefficient identity?

View answer and explanationNo

Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving No. This is coefficient identity, not finite point sampling; the candidate constant is free.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0052/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-candidate · candidate





## 53. Coefficient Eligibility of a Candidate Antiderivative

On all R, f has ascending coefficients [-10, 3, 1, 3], starting with its constant term. The candidate F has ascending rational coefficient pairs [[-11, 1], [-10, 1], [3, 2], [1, 3], [3, 4]], with positive denominators. Is F′=f as a coefficient identity?

View answer and explanationYes

Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving Yes. This is coefficient identity, not finite point sampling; the candidate constant is free.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0053/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-candidate · candidate





## 54. Coefficient Eligibility of a Candidate Antiderivative

On all R, f has ascending coefficients [-9, 4, -1, 4], starting with its constant term. The candidate F has ascending rational coefficient pairs [[-10, 1], [-9, 1], [2, 1], [-1, 3], [2, 1]], with positive denominators. Is F′=f as a coefficient identity?

View answer and explanationNo

Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving No. This is coefficient identity, not finite point sampling; the candidate constant is free.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0054/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-candidate · candidate





## 55. Coefficient Eligibility of a Candidate Antiderivative

On all R, f has ascending coefficients [-8, 5, 0, 1], starting with its constant term. The candidate F has ascending rational coefficient pairs [[-9, 1], [-8, 1], [5, 2], [0, 1], [1, 4]], with positive denominators. Is F′=f as a coefficient identity?

View answer and explanationYes

Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving Yes. This is coefficient identity, not finite point sampling; the candidate constant is free.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0055/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-candidate · candidate





## 56. Coefficient Eligibility of a Candidate Antiderivative

On all R, f has ascending coefficients [-7, 6, 1, 2], starting with its constant term. The candidate F has ascending rational coefficient pairs [[-8, 1], [-7, 1], [3, 1], [1, 3], [3, 2]], with positive denominators. Is F′=f as a coefficient identity?

View answer and explanationNo

Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving No. This is coefficient identity, not finite point sampling; the candidate constant is free.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0056/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-candidate · candidate





## 57. Coefficient Eligibility of a Candidate Antiderivative

On all R, f has ascending coefficients [-6, 7, -1, 3], starting with its constant term. The candidate F has ascending rational coefficient pairs [[-7, 1], [-6, 1], [7, 2], [-1, 3], [3, 4]], with positive denominators. Is F′=f as a coefficient identity?

View answer and explanationYes

Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving Yes. This is coefficient identity, not finite point sampling; the candidate constant is free.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0057/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-candidate · candidate





## 58. Coefficient Eligibility of a Candidate Antiderivative

On all R, f has ascending coefficients [-5, 8, 0, 4], starting with its constant term. The candidate F has ascending rational coefficient pairs [[-6, 1], [-5, 1], [4, 1], [0, 1], [2, 1]], with positive denominators. Is F′=f as a coefficient identity?

View answer and explanationNo

Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving No. This is coefficient identity, not finite point sampling; the candidate constant is free.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0058/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-candidate · candidate





## 59. Coefficient Eligibility of a Candidate Antiderivative

On all R, f has ascending coefficients [-4, 9, 1, 1], starting with its constant term. The candidate F has ascending rational coefficient pairs [[-5, 1], [-4, 1], [9, 2], [1, 3], [1, 4]], with positive denominators. Is F′=f as a coefficient identity?

View answer and explanationYes

Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving Yes. This is coefficient identity, not finite point sampling; the candidate constant is free.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0059/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-candidate · candidate





## 60. Coefficient Eligibility of a Candidate Antiderivative

On all R, f has ascending coefficients [-3, 10, -1, 2], starting with its constant term. The candidate F has ascending rational coefficient pairs [[-4, 1], [-3, 1], [5, 1], [-1, 3], [3, 2]], with positive denominators. Is F′=f as a coefficient identity?

View answer and explanationNo

Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving No. This is coefficient identity, not finite point sampling; the candidate constant is free.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0060/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-candidate · candidate





## 61. Coefficient Eligibility of a Candidate Antiderivative

On all R, f has ascending coefficients [-2, 11, 0, 3], starting with its constant term. The candidate F has ascending rational coefficient pairs [[-3, 1], [-2, 1], [11, 2], [0, 1], [3, 4]], with positive denominators. Is F′=f as a coefficient identity?

View answer and explanationYes

Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving Yes. This is coefficient identity, not finite point sampling; the candidate constant is free.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0061/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-candidate · candidate





## 62. Coefficient Eligibility of a Candidate Antiderivative

On all R, f has ascending coefficients [-1, 12, 1, 4], starting with its constant term. The candidate F has ascending rational coefficient pairs [[-2, 1], [-1, 1], [6, 1], [1, 3], [2, 1]], with positive denominators. Is F′=f as a coefficient identity?

View answer and explanationNo

Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving No. This is coefficient identity, not finite point sampling; the candidate constant is free.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0062/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-candidate · candidate





## 63. Coefficient Eligibility of a Candidate Antiderivative

On all R, f has ascending coefficients [0, 13, -1, 1], starting with its constant term. The candidate F has ascending rational coefficient pairs [[-1, 1], [0, 1], [13, 2], [-1, 3], [1, 4]], with positive denominators. Is F′=f as a coefficient identity?

View answer and explanationYes

Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving Yes. This is coefficient identity, not finite point sampling; the candidate constant is free.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0063/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-candidate · candidate





## 64. Coefficient Eligibility of a Candidate Antiderivative

On all R, f has ascending coefficients [1, 14, 0, 2], starting with its constant term. The candidate F has ascending rational coefficient pairs [[0, 1], [1, 1], [7, 1], [0, 1], [3, 2]], with positive denominators. Is F′=f as a coefficient identity?

View answer and explanationNo

Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving No. This is coefficient identity, not finite point sampling; the candidate constant is free.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0064/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-candidate · candidate





## 65. Coefficient Eligibility of a Candidate Antiderivative

On all R, f has ascending coefficients [2, 15, 1, 3], starting with its constant term. The candidate F has ascending rational coefficient pairs [[1, 1], [2, 1], [15, 2], [1, 3], [3, 4]], with positive denominators. Is F′=f as a coefficient identity?

View answer and explanationYes

Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving Yes. This is coefficient identity, not finite point sampling; the candidate constant is free.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0065/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-candidate · candidate





## 66. Coefficient Eligibility of a Candidate Antiderivative

On all R, f has ascending coefficients [3, 16, -1, 4], starting with its constant term. The candidate F has ascending rational coefficient pairs [[2, 1], [3, 1], [8, 1], [-1, 3], [2, 1]], with positive denominators. Is F′=f as a coefficient identity?

View answer and explanationNo

Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving No. This is coefficient identity, not finite point sampling; the candidate constant is free.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0066/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-candidate · candidate





## 67. Coefficient Eligibility of a Candidate Antiderivative

On all R, f has ascending coefficients [4, 17, 0, 1], starting with its constant term. The candidate F has ascending rational coefficient pairs [[3, 1], [4, 1], [17, 2], [0, 1], [1, 4]], with positive denominators. Is F′=f as a coefficient identity?

View answer and explanationYes

Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving Yes. This is coefficient identity, not finite point sampling; the candidate constant is free.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0067/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-candidate · candidate





## 68. Coefficient Eligibility of a Candidate Antiderivative

On all R, f has ascending coefficients [5, 18, 1, 2], starting with its constant term. The candidate F has ascending rational coefficient pairs [[4, 1], [5, 1], [9, 1], [1, 3], [3, 2]], with positive denominators. Is F′=f as a coefficient identity?

View answer and explanationNo

Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving No. This is coefficient identity, not finite point sampling; the candidate constant is free.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0068/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-candidate · candidate





## 69. Coefficient Eligibility of a Candidate Antiderivative

On all R, f has ascending coefficients [6, 19, -1, 3], starting with its constant term. The candidate F has ascending rational coefficient pairs [[5, 1], [6, 1], [19, 2], [-1, 3], [3, 4]], with positive denominators. Is F′=f as a coefficient identity?

View answer and explanationYes

Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving Yes. This is coefficient identity, not finite point sampling; the candidate constant is free.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0069/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-candidate · candidate





## 70. Coefficient Eligibility of a Candidate Antiderivative

On all R, f has ascending coefficients [7, 20, 0, 4], starting with its constant term. The candidate F has ascending rational coefficient pairs [[6, 1], [7, 1], [10, 1], [0, 1], [2, 1]], with positive denominators. Is F′=f as a coefficient identity?

View answer and explanationNo

Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving No. This is coefficient identity, not finite point sampling; the candidate constant is free.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0070/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-candidate · candidate





## 71. Coefficient Eligibility of a Candidate Antiderivative

On all R, f has ascending coefficients [8, 21, 1, 1], starting with its constant term. The candidate F has ascending rational coefficient pairs [[7, 1], [8, 1], [21, 2], [1, 3], [1, 4]], with positive denominators. Is F′=f as a coefficient identity?

View answer and explanationYes

Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving Yes. This is coefficient identity, not finite point sampling; the candidate constant is free.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0071/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-candidate · candidate





## 72. Coefficient Eligibility of a Candidate Antiderivative

On all R, f has ascending coefficients [9, 22, -1, 2], starting with its constant term. The candidate F has ascending rational coefficient pairs [[8, 1], [9, 1], [11, 1], [-1, 3], [3, 2]], with positive denominators. Is F′=f as a coefficient identity?

View answer and explanationNo

Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving No. This is coefficient identity, not finite point sampling; the candidate constant is free.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0072/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-candidate · candidate





## 73. Coefficient Eligibility of a Candidate Antiderivative

On all R, f has ascending coefficients [10, 23, 0, 3], starting with its constant term. The candidate F has ascending rational coefficient pairs [[9, 1], [10, 1], [23, 2], [0, 1], [3, 4]], with positive denominators. Is F′=f as a coefficient identity?

View answer and explanationYes

Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving Yes. This is coefficient identity, not finite point sampling; the candidate constant is free.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0073/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-candidate · candidate





## 74. Coefficient Eligibility of a Candidate Antiderivative

On all R, f has ascending coefficients [11, 24, 1, 4], starting with its constant term. The candidate F has ascending rational coefficient pairs [[10, 1], [11, 1], [12, 1], [1, 3], [2, 1]], with positive denominators. Is F′=f as a coefficient identity?

View answer and explanationNo

Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving No. This is coefficient identity, not finite point sampling; the candidate constant is free.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0074/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-candidate · candidate





## 75. Coefficient Eligibility of a Candidate Antiderivative

On all R, f has ascending coefficients [12, 25, -1, 1], starting with its constant term. The candidate F has ascending rational coefficient pairs [[11, 1], [12, 1], [25, 2], [-1, 3], [1, 4]], with positive denominators. Is F′=f as a coefficient identity?

View answer and explanationYes

Differentiate each positive-degree rational coefficient and compare the full normalized coefficient lists, giving Yes. This is coefficient identity, not finite point sampling; the candidate constant is free.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0075/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-candidate · candidate





## 76. Antiderivative Value Retaining the Actual Constant

On all R, f has ascending coefficients [1, 2], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(0)+(0), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation0/1

Evaluate the rational primitive exactly at 0, then add C=0, obtaining 0/1. A constant irrelevant to differentiation remains relevant to this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0076/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-shifted-value · candidate





## 77. Antiderivative Value Retaining the Actual Constant

On all R, f has ascending coefficients [0], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(3)+(1), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation1/1

Evaluate the rational primitive exactly at 3, then add C=1, obtaining 1/1. A constant irrelevant to differentiation remains relevant to this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0077/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-shifted-value · candidate





## 78. Antiderivative Value Retaining the Actual Constant

On all R, f has ascending coefficients [-10, 3, 1, 3], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(3)+(-10), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation173/4

Evaluate the rational primitive exactly at 3, then add C=-10, obtaining 173/4. A constant irrelevant to differentiation remains relevant to this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0078/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-shifted-value · candidate





## 79. Antiderivative Value Retaining the Actual Constant

On all R, f has ascending coefficients [-9, 4, -1, 4], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(-1)+(-9), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation10/3

Evaluate the rational primitive exactly at -1, then add C=-9, obtaining 10/3. A constant irrelevant to differentiation remains relevant to this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0079/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-shifted-value · candidate





## 80. Antiderivative Value Retaining the Actual Constant

On all R, f has ascending coefficients [-8, 5, 0, 1], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(2)+(-8), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-10/1

Evaluate the rational primitive exactly at 2, then add C=-8, obtaining -10/1. A constant irrelevant to differentiation remains relevant to this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0080/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-shifted-value · candidate





## 81. Antiderivative Value Retaining the Actual Constant

On all R, f has ascending coefficients [-7, 6, 1, 2], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(-3)+(-7), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation145/2

Evaluate the rational primitive exactly at -3, then add C=-7, obtaining 145/2. A constant irrelevant to differentiation remains relevant to this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0081/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-shifted-value · candidate





## 82. Antiderivative Value Retaining the Actual Constant

On all R, f has ascending coefficients [-6, 7, -1, 3], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(1)+(-6), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation-97/12

Evaluate the rational primitive exactly at 1, then add C=-6, obtaining -97/12. A constant irrelevant to differentiation remains relevant to this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0082/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-shifted-value · candidate





## 83. Antiderivative Value Retaining the Actual Constant

On all R, f has ascending coefficients [-5, 8, 0, 4], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(-2)+(-5), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation37/1

Evaluate the rational primitive exactly at -2, then add C=-5, obtaining 37/1. A constant irrelevant to differentiation remains relevant to this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0083/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-shifted-value · candidate





## 84. Antiderivative Value Retaining the Actual Constant

On all R, f has ascending coefficients [-4, 9, 1, 1], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(3)+(-4), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation215/4

Evaluate the rational primitive exactly at 3, then add C=-4, obtaining 215/4. A constant irrelevant to differentiation remains relevant to this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0084/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-shifted-value · candidate





## 85. Antiderivative Value Retaining the Actual Constant

On all R, f has ascending coefficients [-3, 10, -1, 2], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(-1)+(-3), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation35/6

Evaluate the rational primitive exactly at -1, then add C=-3, obtaining 35/6. A constant irrelevant to differentiation remains relevant to this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0085/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-shifted-value · candidate





## 86. Antiderivative Value Retaining the Actual Constant

On all R, f has ascending coefficients [-2, 11, 0, 3], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(2)+(-2), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation28/1

Evaluate the rational primitive exactly at 2, then add C=-2, obtaining 28/1. A constant irrelevant to differentiation remains relevant to this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0086/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-shifted-value · candidate





## 87. Antiderivative Value Retaining the Actual Constant

On all R, f has ascending coefficients [-1, 12, 1, 4], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(-3)+(-1), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation128/1

Evaluate the rational primitive exactly at -3, then add C=-1, obtaining 128/1. A constant irrelevant to differentiation remains relevant to this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0087/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-shifted-value · candidate





## 88. Antiderivative Value Retaining the Actual Constant

On all R, f has ascending coefficients [0, 13, -1, 1], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(1)+(0), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation77/12

Evaluate the rational primitive exactly at 1, then add C=0, obtaining 77/12. A constant irrelevant to differentiation remains relevant to this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0088/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-shifted-value · candidate





## 89. Antiderivative Value Retaining the Actual Constant

On all R, f has ascending coefficients [1, 14, 0, 2], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(-2)+(1), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation35/1

Evaluate the rational primitive exactly at -2, then add C=1, obtaining 35/1. A constant irrelevant to differentiation remains relevant to this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0089/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-shifted-value · candidate





## 90. Antiderivative Value Retaining the Actual Constant

On all R, f has ascending coefficients [2, 15, 1, 3], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(3)+(2), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation581/4

Evaluate the rational primitive exactly at 3, then add C=2, obtaining 581/4. A constant irrelevant to differentiation remains relevant to this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0090/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-shifted-value · candidate





## 91. Antiderivative Value Retaining the Actual Constant

On all R, f has ascending coefficients [3, 16, -1, 4], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(-1)+(3), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation28/3

Evaluate the rational primitive exactly at -1, then add C=3, obtaining 28/3. A constant irrelevant to differentiation remains relevant to this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0091/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-shifted-value · candidate





## 92. Antiderivative Value Retaining the Actual Constant

On all R, f has ascending coefficients [4, 17, 0, 1], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(2)+(4), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation50/1

Evaluate the rational primitive exactly at 2, then add C=4, obtaining 50/1. A constant irrelevant to differentiation remains relevant to this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0092/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-shifted-value · candidate





## 93. Antiderivative Value Retaining the Actual Constant

On all R, f has ascending coefficients [5, 18, 1, 2], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(-3)+(5), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation205/2

Evaluate the rational primitive exactly at -3, then add C=5, obtaining 205/2. A constant irrelevant to differentiation remains relevant to this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0093/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-shifted-value · candidate





## 94. Antiderivative Value Retaining the Actual Constant

On all R, f has ascending coefficients [6, 19, -1, 3], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(1)+(6), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation263/12

Evaluate the rational primitive exactly at 1, then add C=6, obtaining 263/12. A constant irrelevant to differentiation remains relevant to this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0094/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-shifted-value · candidate





## 95. Antiderivative Value Retaining the Actual Constant

On all R, f has ascending coefficients [7, 20, 0, 4], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(-2)+(7), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation49/1

Evaluate the rational primitive exactly at -2, then add C=7, obtaining 49/1. A constant irrelevant to differentiation remains relevant to this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0095/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-shifted-value · candidate





## 96. Antiderivative Value Retaining the Actual Constant

On all R, f has ascending coefficients [8, 21, 1, 1], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(3)+(8), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation623/4

Evaluate the rational primitive exactly at 3, then add C=8, obtaining 623/4. A constant irrelevant to differentiation remains relevant to this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0096/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-shifted-value · candidate





## 97. Antiderivative Value Retaining the Actual Constant

On all R, f has ascending coefficients [9, 22, -1, 2], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(-1)+(9), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation71/6

Evaluate the rational primitive exactly at -1, then add C=9, obtaining 71/6. A constant irrelevant to differentiation remains relevant to this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0097/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-shifted-value · candidate





## 98. Antiderivative Value Retaining the Actual Constant

On all R, f has ascending coefficients [10, 23, 0, 3], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(2)+(10), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation88/1

Evaluate the rational primitive exactly at 2, then add C=10, obtaining 88/1. A constant irrelevant to differentiation remains relevant to this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0098/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-shifted-value · candidate





## 99. Antiderivative Value Retaining the Actual Constant

On all R, f has ascending coefficients [11, 24, 1, 4], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(-3)+(11), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation158/1

Evaluate the rational primitive exactly at -3, then add C=11, obtaining 158/1. A constant irrelevant to differentiation remains relevant to this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0099/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-shifted-value · candidate





## 100. Antiderivative Value Retaining the Actual Constant

On all R, f has ascending coefficients [12, 25, -1, 1], starting with its constant term. Let F0 be its zero-constant primitive. Find F0(1)+(12), retaining the actual additive constant. Give a reduced fraction with positive denominator, retaining /1 for integers.

View answer and explanation437/12

Evaluate the rational primitive exactly at 1, then add C=12, obtaining 437/12. A constant irrelevant to differentiation remains relevant to this value.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261007-advance-20261008-r06-antiderivative-0100/)

Polynomial antiderivatives and the choice of constant · qf-antiderivative-shifted-value · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261007-advance-20261008-r06-antiderivative-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261007-advance-20261008-r06-antiderivative-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261007-advance-20261008-r06-antiderivative-foundations/validation-report.json)