Polynomial Antiderivative and Constant Foundations
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
No
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
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 explanation
Yes
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
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 explanation
0/1
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
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 explanation
1/1
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
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 explanation
173/4
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
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 explanation
10/3
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
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
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 explanation
145/2
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
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
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 explanation
37/1
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
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 explanation
215/4
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
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 explanation
35/6
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
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 explanation
28/1
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
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 explanation
128/1
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
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 explanation
77/12
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
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 explanation
35/1
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
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.
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581/4
Evaluate the rational primitive exactly at 3, then add C=2, obtaining 581/4. A constant irrelevant to differentiation remains relevant to this value.
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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.
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28/3
Evaluate the rational primitive exactly at -1, then add C=3, obtaining 28/3. A constant irrelevant to differentiation remains relevant to this value.
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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.
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50/1
Evaluate the rational primitive exactly at 2, then add C=4, obtaining 50/1. A constant irrelevant to differentiation remains relevant to this value.
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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.
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205/2
Evaluate the rational primitive exactly at -3, then add C=5, obtaining 205/2. A constant irrelevant to differentiation remains relevant to this value.
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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.
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263/12
Evaluate the rational primitive exactly at 1, then add C=6, obtaining 263/12. A constant irrelevant to differentiation remains relevant to this value.
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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.
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49/1
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
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.
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623/4
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
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.
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71/6
Evaluate the rational primitive exactly at -1, then add C=9, obtaining 71/6. A constant irrelevant to differentiation remains relevant to this value.
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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.
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88/1
Evaluate the rational primitive exactly at 2, then add C=10, obtaining 88/1. A constant irrelevant to differentiation remains relevant to this value.
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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.
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158/1
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
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.
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437/12
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
Polynomial antiderivatives and the choice of constant · qf-antiderivative-shifted-value · candidate
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