Primitive Coefficient, Constant, and Anchor Depth
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batch-20261007-advance-20261009-r05-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. Specified Coefficient of the Zero-Constant Primitive
On all R, f has ascending coefficients [0], constant term first. For its zero-constant primitive F0, find the coefficient of x^1. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
0/1
Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain 0/1. The order is ascending and the representative constant is zero.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-coefficient · candidate
2. Specified Coefficient of the Zero-Constant Primitive
On all R, f has ascending coefficients [2], constant term first. For its zero-constant primitive F0, find the coefficient of x^4. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
0/1
Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain 0/1. The order is ascending and the representative constant is zero.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-coefficient · candidate
3. Specified Coefficient of the Zero-Constant Primitive
On all R, f has ascending coefficients [-10, 3, -4, 5], constant term first. For its zero-constant primitive F0, find the coefficient of x^3. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-4/3
Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain -4/3. The order is ascending and the representative constant is zero.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-coefficient · candidate
4. Specified Coefficient of the Zero-Constant Primitive
On all R, f has ascending coefficients [-9, 4, -5, -6], constant term first. For its zero-constant primitive F0, find the coefficient of x^4. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-3/2
Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain -3/2. The order is ascending and the representative constant is zero.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-coefficient · candidate
5. Specified Coefficient of the Zero-Constant Primitive
On all R, f has ascending coefficients [-8, 5, -6, 7], constant term first. For its zero-constant primitive F0, find the coefficient of x^1. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-8/1
Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain -8/1. The order is ascending and the representative constant is zero.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-coefficient · candidate
6. Specified Coefficient of the Zero-Constant Primitive
On all R, f has ascending coefficients [-7, 6, -7, -8], constant term first. For its zero-constant primitive F0, find the coefficient of x^2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
3/1
Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain 3/1. The order is ascending and the representative constant is zero.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-coefficient · candidate
7. Specified Coefficient of the Zero-Constant Primitive
On all R, f has ascending coefficients [-6, 7, -8, 9], constant term first. For its zero-constant primitive F0, find the coefficient of x^3. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-8/3
Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain -8/3. The order is ascending and the representative constant is zero.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-coefficient · candidate
8. Specified Coefficient of the Zero-Constant Primitive
On all R, f has ascending coefficients [-5, 8, -9, -10], constant term first. For its zero-constant primitive F0, find the coefficient of x^4. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-5/2
Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain -5/2. The order is ascending and the representative constant is zero.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-coefficient · candidate
9. Specified Coefficient of the Zero-Constant Primitive
On all R, f has ascending coefficients [-4, 9, -10, 11], constant term first. For its zero-constant primitive F0, find the coefficient of x^1. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-4/1
Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain -4/1. The order is ascending and the representative constant is zero.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-coefficient · candidate
10. Specified Coefficient of the Zero-Constant Primitive
On all R, f has ascending coefficients [-3, 10, -11, -12], constant term first. For its zero-constant primitive F0, find the coefficient of x^2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
5/1
Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain 5/1. The order is ascending and the representative constant is zero.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-coefficient · candidate
11. Specified Coefficient of the Zero-Constant Primitive
On all R, f has ascending coefficients [-2, 11, -12, 13], constant term first. For its zero-constant primitive F0, find the coefficient of x^3. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-4/1
Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain -4/1. The order is ascending and the representative constant is zero.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-coefficient · candidate
12. Specified Coefficient of the Zero-Constant Primitive
On all R, f has ascending coefficients [-1, 12, -13, -14], constant term first. For its zero-constant primitive F0, find the coefficient of x^4. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-7/2
Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain -7/2. The order is ascending and the representative constant is zero.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-coefficient · candidate
13. Specified Coefficient of the Zero-Constant Primitive
On all R, f has ascending coefficients [1, 14, -15, -16], constant term first. For its zero-constant primitive F0, find the coefficient of x^2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
7/1
Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain 7/1. The order is ascending and the representative constant is zero.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-coefficient · candidate
14. Specified Coefficient of the Zero-Constant Primitive
On all R, f has ascending coefficients [2, 15, -16, 17], constant term first. For its zero-constant primitive F0, find the coefficient of x^3. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-16/3
Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain -16/3. The order is ascending and the representative constant is zero.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-coefficient · candidate
15. Specified Coefficient of the Zero-Constant Primitive
On all R, f has ascending coefficients [3, 16, -17, -18], constant term first. For its zero-constant primitive F0, find the coefficient of x^4. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-9/2
Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain -9/2. The order is ascending and the representative constant is zero.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-coefficient · candidate
16. Specified Coefficient of the Zero-Constant Primitive
On all R, f has ascending coefficients [4, 17, -18, 19], constant term first. For its zero-constant primitive F0, find the coefficient of x^1. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
4/1
Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain 4/1. The order is ascending and the representative constant is zero.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-coefficient · candidate
17. Specified Coefficient of the Zero-Constant Primitive
On all R, f has ascending coefficients [5, 18, -19, -20], constant term first. For its zero-constant primitive F0, find the coefficient of x^2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
9/1
Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain 9/1. The order is ascending and the representative constant is zero.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-coefficient · candidate
18. Specified Coefficient of the Zero-Constant Primitive
On all R, f has ascending coefficients [6, 19, -20, 21], constant term first. For its zero-constant primitive F0, find the coefficient of x^3. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-20/3
Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain -20/3. The order is ascending and the representative constant is zero.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-coefficient · candidate
19. Specified Coefficient of the Zero-Constant Primitive
On all R, f has ascending coefficients [7, 20, -21, -22], constant term first. For its zero-constant primitive F0, find the coefficient of x^4. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-11/2
Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain -11/2. The order is ascending and the representative constant is zero.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-coefficient · candidate
20. Specified Coefficient of the Zero-Constant Primitive
On all R, f has ascending coefficients [8, 21, -22, 23], constant term first. For its zero-constant primitive F0, find the coefficient of x^1. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
8/1
Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain 8/1. The order is ascending and the representative constant is zero.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-coefficient · candidate
21. Specified Coefficient of the Zero-Constant Primitive
On all R, f has ascending coefficients [9, 22, -23, -24], constant term first. For its zero-constant primitive F0, find the coefficient of x^2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
11/1
Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain 11/1. The order is ascending and the representative constant is zero.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-coefficient · candidate
22. Specified Coefficient of the Zero-Constant Primitive
On all R, f has ascending coefficients [10, 23, -24, 25], constant term first. For its zero-constant primitive F0, find the coefficient of x^3. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-8/1
Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain -8/1. The order is ascending and the representative constant is zero.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-coefficient · candidate
23. Specified Coefficient of the Zero-Constant Primitive
On all R, f has ascending coefficients [11, 24, -25, -26], constant term first. For its zero-constant primitive F0, find the coefficient of x^4. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-13/2
Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain -13/2. The order is ascending and the representative constant is zero.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-coefficient · candidate
24. Specified Coefficient of the Zero-Constant Primitive
On all R, f has ascending coefficients [12, 25, -26, 27], constant term first. For its zero-constant primitive F0, find the coefficient of x^1. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
12/1
Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain 12/1. The order is ascending and the representative constant is zero.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-coefficient · candidate
25. Specified Coefficient of the Zero-Constant Primitive
On all R, f has ascending coefficients [13, 26, -27, -28], constant term first. For its zero-constant primitive F0, find the coefficient of x^2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
13/1
Divide the input coefficient at power k−1 by k, using zero for an absent coefficient, to obtain 13/1. The order is ascending and the representative constant is zero.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-coefficient · candidate
26. Difference of Constants for a Shared Primitive
Two full-real polynomial primitives share F0: F1=F0+(-12) and F2=F0+(0). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-12/1
The shared primitive cancels, leaving C1−C2=-12/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-constant-difference · candidate
27. Difference of Constants for a Shared Primitive
Two full-real polynomial primitives share F0: F1=F0+(-11) and F2=F0+(-1). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-10/1
The shared primitive cancels, leaving C1−C2=-10/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-constant-difference · candidate
28. Difference of Constants for a Shared Primitive
Two full-real polynomial primitives share F0: F1=F0+(-10) and F2=F0+(-2). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-8/1
The shared primitive cancels, leaving C1−C2=-8/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-constant-difference · candidate
29. Difference of Constants for a Shared Primitive
Two full-real polynomial primitives share F0: F1=F0+(-9) and F2=F0+(-3). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-6/1
The shared primitive cancels, leaving C1−C2=-6/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-constant-difference · candidate
30. Difference of Constants for a Shared Primitive
Two full-real polynomial primitives share F0: F1=F0+(-8) and F2=F0+(-4). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-4/1
The shared primitive cancels, leaving C1−C2=-4/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-constant-difference · candidate
31. Difference of Constants for a Shared Primitive
Two full-real polynomial primitives share F0: F1=F0+(-7) and F2=F0+(-5). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-2/1
The shared primitive cancels, leaving C1−C2=-2/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-constant-difference · candidate
32. Difference of Constants for a Shared Primitive
Two full-real polynomial primitives share F0: F1=F0+(-6) and F2=F0+(-6). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
0/1
The shared primitive cancels, leaving C1−C2=0/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-constant-difference · candidate
33. Difference of Constants for a Shared Primitive
Two full-real polynomial primitives share F0: F1=F0+(-5) and F2=F0+(-7). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
2/1
The shared primitive cancels, leaving C1−C2=2/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-constant-difference · candidate
34. Difference of Constants for a Shared Primitive
Two full-real polynomial primitives share F0: F1=F0+(-4) and F2=F0+(-8). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
4/1
The shared primitive cancels, leaving C1−C2=4/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-constant-difference · candidate
35. Difference of Constants for a Shared Primitive
Two full-real polynomial primitives share F0: F1=F0+(-3) and F2=F0+(-9). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
6/1
The shared primitive cancels, leaving C1−C2=6/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-constant-difference · candidate
36. Difference of Constants for a Shared Primitive
Two full-real polynomial primitives share F0: F1=F0+(-2) and F2=F0+(-10). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
8/1
The shared primitive cancels, leaving C1−C2=8/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-constant-difference · candidate
37. Difference of Constants for a Shared Primitive
Two full-real polynomial primitives share F0: F1=F0+(-1) and F2=F0+(-11). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
10/1
The shared primitive cancels, leaving C1−C2=10/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-constant-difference · candidate
38. Difference of Constants for a Shared Primitive
Two full-real polynomial primitives share F0: F1=F0+(0) and F2=F0+(-12). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
12/1
The shared primitive cancels, leaving C1−C2=12/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-constant-difference · candidate
39. Difference of Constants for a Shared Primitive
Two full-real polynomial primitives share F0: F1=F0+(1) and F2=F0+(-13). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
14/1
The shared primitive cancels, leaving C1−C2=14/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-constant-difference · candidate
40. Difference of Constants for a Shared Primitive
Two full-real polynomial primitives share F0: F1=F0+(2) and F2=F0+(-14). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
16/1
The shared primitive cancels, leaving C1−C2=16/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-constant-difference · candidate
41. Difference of Constants for a Shared Primitive
Two full-real polynomial primitives share F0: F1=F0+(3) and F2=F0+(-15). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
18/1
The shared primitive cancels, leaving C1−C2=18/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-constant-difference · candidate
42. Difference of Constants for a Shared Primitive
Two full-real polynomial primitives share F0: F1=F0+(4) and F2=F0+(-16). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
20/1
The shared primitive cancels, leaving C1−C2=20/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-constant-difference · candidate
43. Difference of Constants for a Shared Primitive
Two full-real polynomial primitives share F0: F1=F0+(5) and F2=F0+(-17). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
22/1
The shared primitive cancels, leaving C1−C2=22/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-constant-difference · candidate
44. Difference of Constants for a Shared Primitive
Two full-real polynomial primitives share F0: F1=F0+(6) and F2=F0+(-18). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
24/1
The shared primitive cancels, leaving C1−C2=24/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-constant-difference · candidate
45. Difference of Constants for a Shared Primitive
Two full-real polynomial primitives share F0: F1=F0+(7) and F2=F0+(-19). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
26/1
The shared primitive cancels, leaving C1−C2=26/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-constant-difference · candidate
46. Difference of Constants for a Shared Primitive
Two full-real polynomial primitives share F0: F1=F0+(8) and F2=F0+(-20). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
28/1
The shared primitive cancels, leaving C1−C2=28/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-constant-difference · candidate
47. Difference of Constants for a Shared Primitive
Two full-real polynomial primitives share F0: F1=F0+(9) and F2=F0+(-21). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
30/1
The shared primitive cancels, leaving C1−C2=30/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-constant-difference · candidate
48. Difference of Constants for a Shared Primitive
Two full-real polynomial primitives share F0: F1=F0+(10) and F2=F0+(-22). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
32/1
The shared primitive cancels, leaving C1−C2=32/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-constant-difference · candidate
49. Difference of Constants for a Shared Primitive
Two full-real polynomial primitives share F0: F1=F0+(11) and F2=F0+(-23). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
34/1
The shared primitive cancels, leaving C1−C2=34/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-constant-difference · candidate
50. Difference of Constants for a Shared Primitive
Two full-real polynomial primitives share F0: F1=F0+(12) and F2=F0+(-24). Find the constant F1−F2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
36/1
The shared primitive cancels, leaving C1−C2=36/1. No irrelevant polynomial or point is introduced, and this scoped algebraic identity does not establish a theorem on disconnected domains.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-constant-difference · candidate
51. Primitive Constant Selected by an Explicit Anchor
On all R, f has ascending coefficients [1, 1], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(0)=1/2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
1/2
Compute C=V−F0(a) exactly to obtain 1/2. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-anchor-constant · candidate
52. Primitive Constant Selected by an Explicit Anchor
On all R, f has ascending coefficients [0], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(3)=-2/1. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-2/1
Compute C=V−F0(a) exactly to obtain -2/1. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-anchor-constant · candidate
53. Primitive Constant Selected by an Explicit Anchor
On all R, f has ascending coefficients [-10, 3, 1, 3], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-3)=-10/3. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-1183/12
Compute C=V−F0(a) exactly to obtain -1183/12. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-anchor-constant · candidate
54. Primitive Constant Selected by an Explicit Anchor
On all R, f has ascending coefficients [-9, 4, -1, 4], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-4)=-9/1. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-1063/3
Compute C=V−F0(a) exactly to obtain -1063/3. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-anchor-constant · candidate
55. Primitive Constant Selected by an Explicit Anchor
On all R, f has ascending coefficients [-8, 5, 0, 1], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-1)=-8/2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-59/4
Compute C=V−F0(a) exactly to obtain -59/4. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-anchor-constant · candidate
56. Primitive Constant Selected by an Explicit Anchor
On all R, f has ascending coefficients [-7, 6, 1, 2], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-2)=-7/3. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-101/3
Compute C=V−F0(a) exactly to obtain -101/3. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-anchor-constant · candidate
57. Primitive Constant Selected by an Explicit Anchor
On all R, f has ascending coefficients [-6, 7, -1, 3], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-3)=-6/1. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-501/4
Compute C=V−F0(a) exactly to obtain -501/4. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-anchor-constant · candidate
58. Primitive Constant Selected by an Explicit Anchor
On all R, f has ascending coefficients [-5, 8, 0, 4], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-4)=-5/2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-685/2
Compute C=V−F0(a) exactly to obtain -685/2. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-anchor-constant · candidate
59. Primitive Constant Selected by an Explicit Anchor
On all R, f has ascending coefficients [-4, 9, 1, 1], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-1)=-4/3. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-39/4
Compute C=V−F0(a) exactly to obtain -39/4. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-anchor-constant · candidate
60. Primitive Constant Selected by an Explicit Anchor
On all R, f has ascending coefficients [-3, 10, -1, 2], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-2)=-3/1. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-119/3
Compute C=V−F0(a) exactly to obtain -119/3. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-anchor-constant · candidate
61. Primitive Constant Selected by an Explicit Anchor
On all R, f has ascending coefficients [-2, 11, 0, 3], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-3)=-2/2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-469/4
Compute C=V−F0(a) exactly to obtain -469/4. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-anchor-constant · candidate
62. Primitive Constant Selected by an Explicit Anchor
On all R, f has ascending coefficients [-1, 12, 1, 4], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-4)=-1/3. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-335/1
Compute C=V−F0(a) exactly to obtain -335/1. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-anchor-constant · candidate
63. Primitive Constant Selected by an Explicit Anchor
On all R, f has ascending coefficients [0, 13, -1, 1], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-1)=0/1. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-85/12
Compute C=V−F0(a) exactly to obtain -85/12. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-anchor-constant · candidate
64. Primitive Constant Selected by an Explicit Anchor
On all R, f has ascending coefficients [1, 14, 0, 2], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-2)=1/2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-67/2
Compute C=V−F0(a) exactly to obtain -67/2. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-anchor-constant · candidate
65. Primitive Constant Selected by an Explicit Anchor
On all R, f has ascending coefficients [2, 15, 1, 3], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-3)=2/3. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-1351/12
Compute C=V−F0(a) exactly to obtain -1351/12. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-anchor-constant · candidate
66. Primitive Constant Selected by an Explicit Anchor
On all R, f has ascending coefficients [3, 16, -1, 4], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-4)=3/1. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-1171/3
Compute C=V−F0(a) exactly to obtain -1171/3. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-anchor-constant · candidate
67. Primitive Constant Selected by an Explicit Anchor
On all R, f has ascending coefficients [4, 17, 0, 1], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-1)=4/2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-11/4
Compute C=V−F0(a) exactly to obtain -11/4. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-anchor-constant · candidate
68. Primitive Constant Selected by an Explicit Anchor
On all R, f has ascending coefficients [5, 18, 1, 2], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-2)=5/3. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-89/3
Compute C=V−F0(a) exactly to obtain -89/3. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-anchor-constant · candidate
69. Primitive Constant Selected by an Explicit Anchor
On all R, f has ascending coefficients [6, 19, -1, 3], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-3)=6/1. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-525/4
Compute C=V−F0(a) exactly to obtain -525/4. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-anchor-constant · candidate
70. Primitive Constant Selected by an Explicit Anchor
On all R, f has ascending coefficients [7, 20, 0, 4], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-4)=7/2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-769/2
Compute C=V−F0(a) exactly to obtain -769/2. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-anchor-constant · candidate
71. Primitive Constant Selected by an Explicit Anchor
On all R, f has ascending coefficients [8, 21, 1, 1], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-1)=8/3. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
1/4
Compute C=V−F0(a) exactly to obtain 1/4. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-anchor-constant · candidate
72. Primitive Constant Selected by an Explicit Anchor
On all R, f has ascending coefficients [9, 22, -1, 2], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-2)=9/1. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-83/3
Compute C=V−F0(a) exactly to obtain -83/3. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-anchor-constant · candidate
73. Primitive Constant Selected by an Explicit Anchor
On all R, f has ascending coefficients [10, 23, 0, 3], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-3)=10/2. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-517/4
Compute C=V−F0(a) exactly to obtain -517/4. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-anchor-constant · candidate
74. Primitive Constant Selected by an Explicit Anchor
On all R, f has ascending coefficients [11, 24, 1, 4], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-4)=11/3. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
-379/1
Compute C=V−F0(a) exactly to obtain -379/1. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-anchor-constant · candidate
75. Primitive Constant Selected by an Explicit Anchor
On all R, f has ascending coefficients [12, 25, -1, 1], constant term first. Let F=F0+C with zero-constant primitive F0. Choose C so that F(-1)=12/1. Use reduced positive-denominator fractions, retaining /1 for integers.
View answer and explanation
131/12
Compute C=V−F0(a) exactly to obtain 131/12. Retain the supplied anchor and target value; an additive constant matters for evaluation even though it disappears under differentiation.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-anchor-constant · candidate
76. Derivative Coefficients of a Rational Candidate
On all R, F has ascending rational coefficients [7/1]. Give its complete normalized ascending derivative coefficients as reduced fractions.
View answer and explanation
[0/1]
Differentiate coefficientwise to obtain [0/1]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-candidate-derivative · candidate
77. Derivative Coefficients of a Rational Candidate
On all R, F has ascending rational coefficients [3/1, 2/3]. Give its complete normalized ascending derivative coefficients as reduced fractions.
View answer and explanation
[2/3]
Differentiate coefficientwise to obtain [2/3]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-candidate-derivative · candidate
78. Derivative Coefficients of a Rational Candidate
On all R, F has ascending rational coefficients [-10/1, 3/2, -1/3, 1/1, 2/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.
View answer and explanation
[3/2, -2/3, 3/1, 8/5]
Differentiate coefficientwise to obtain [3/2, -2/3, 3/1, 8/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-candidate-derivative · candidate
79. Derivative Coefficients of a Rational Candidate
On all R, F has ascending rational coefficients [-9/1, 2/1, -2/3, 5/4, -3/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.
View answer and explanation
[2/1, -4/3, 15/4, -12/5]
Differentiate coefficientwise to obtain [2/1, -4/3, 15/4, -12/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-candidate-derivative · candidate
80. Derivative Coefficients of a Rational Candidate
On all R, F has ascending rational coefficients [-8/1, 5/2, -1/1, 3/2, 4/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.
View answer and explanation
[5/2, -2/1, 9/2, 16/5]
Differentiate coefficientwise to obtain [5/2, -2/1, 9/2, 16/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-candidate-derivative · candidate
81. Derivative Coefficients of a Rational Candidate
On all R, F has ascending rational coefficients [-7/1, 3/1, -4/3, 7/4, -1/1]. Give its complete normalized ascending derivative coefficients as reduced fractions.
View answer and explanation
[3/1, -8/3, 21/4, -4/1]
Differentiate coefficientwise to obtain [3/1, -8/3, 21/4, -4/1]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-candidate-derivative · candidate
82. Derivative Coefficients of a Rational Candidate
On all R, F has ascending rational coefficients [-6/1, 7/2, -5/3, 2/1, 6/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.
View answer and explanation
[7/2, -10/3, 6/1, 24/5]
Differentiate coefficientwise to obtain [7/2, -10/3, 6/1, 24/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-candidate-derivative · candidate
83. Derivative Coefficients of a Rational Candidate
On all R, F has ascending rational coefficients [-5/1, 4/1, -2/1, 9/4, -7/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.
View answer and explanation
[4/1, -4/1, 27/4, -28/5]
Differentiate coefficientwise to obtain [4/1, -4/1, 27/4, -28/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-candidate-derivative · candidate
84. Derivative Coefficients of a Rational Candidate
On all R, F has ascending rational coefficients [-4/1, 9/2, -7/3, 5/2, 8/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.
View answer and explanation
[9/2, -14/3, 15/2, 32/5]
Differentiate coefficientwise to obtain [9/2, -14/3, 15/2, 32/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-candidate-derivative · candidate
85. Derivative Coefficients of a Rational Candidate
On all R, F has ascending rational coefficients [-3/1, 5/1, -8/3, 11/4, -9/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.
View answer and explanation
[5/1, -16/3, 33/4, -36/5]
Differentiate coefficientwise to obtain [5/1, -16/3, 33/4, -36/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-candidate-derivative · candidate
86. Derivative Coefficients of a Rational Candidate
On all R, F has ascending rational coefficients [-2/1, 11/2, -3/1, 3/1, 2/1]. Give its complete normalized ascending derivative coefficients as reduced fractions.
View answer and explanation
[11/2, -6/1, 9/1, 8/1]
Differentiate coefficientwise to obtain [11/2, -6/1, 9/1, 8/1]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-candidate-derivative · candidate
87. Derivative Coefficients of a Rational Candidate
On all R, F has ascending rational coefficients [-1/1, 6/1, -10/3, 13/4, -11/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.
View answer and explanation
[6/1, -20/3, 39/4, -44/5]
Differentiate coefficientwise to obtain [6/1, -20/3, 39/4, -44/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-candidate-derivative · candidate
88. Derivative Coefficients of a Rational Candidate
On all R, F has ascending rational coefficients [0/1, 13/2, -11/3, 7/2, 12/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.
View answer and explanation
[13/2, -22/3, 21/2, 48/5]
Differentiate coefficientwise to obtain [13/2, -22/3, 21/2, 48/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-candidate-derivative · candidate
89. Derivative Coefficients of a Rational Candidate
On all R, F has ascending rational coefficients [1/1, 7/1, -4/1, 15/4, -13/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.
View answer and explanation
[7/1, -8/1, 45/4, -52/5]
Differentiate coefficientwise to obtain [7/1, -8/1, 45/4, -52/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-candidate-derivative · candidate
90. Derivative Coefficients of a Rational Candidate
On all R, F has ascending rational coefficients [2/1, 15/2, -13/3, 4/1, 14/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.
View answer and explanation
[15/2, -26/3, 12/1, 56/5]
Differentiate coefficientwise to obtain [15/2, -26/3, 12/1, 56/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-candidate-derivative · candidate
91. Derivative Coefficients of a Rational Candidate
On all R, F has ascending rational coefficients [3/1, 8/1, -14/3, 17/4, -3/1]. Give its complete normalized ascending derivative coefficients as reduced fractions.
View answer and explanation
[8/1, -28/3, 51/4, -12/1]
Differentiate coefficientwise to obtain [8/1, -28/3, 51/4, -12/1]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-candidate-derivative · candidate
92. Derivative Coefficients of a Rational Candidate
On all R, F has ascending rational coefficients [4/1, 17/2, -5/1, 9/2, 16/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.
View answer and explanation
[17/2, -10/1, 27/2, 64/5]
Differentiate coefficientwise to obtain [17/2, -10/1, 27/2, 64/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-candidate-derivative · candidate
93. Derivative Coefficients of a Rational Candidate
On all R, F has ascending rational coefficients [5/1, 9/1, -16/3, 19/4, -17/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.
View answer and explanation
[9/1, -32/3, 57/4, -68/5]
Differentiate coefficientwise to obtain [9/1, -32/3, 57/4, -68/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-candidate-derivative · candidate
94. Derivative Coefficients of a Rational Candidate
On all R, F has ascending rational coefficients [6/1, 19/2, -17/3, 5/1, 18/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.
View answer and explanation
[19/2, -34/3, 15/1, 72/5]
Differentiate coefficientwise to obtain [19/2, -34/3, 15/1, 72/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-candidate-derivative · candidate
95. Derivative Coefficients of a Rational Candidate
On all R, F has ascending rational coefficients [7/1, 10/1, -6/1, 21/4, -19/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.
View answer and explanation
[10/1, -12/1, 63/4, -76/5]
Differentiate coefficientwise to obtain [10/1, -12/1, 63/4, -76/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-candidate-derivative · candidate
96. Derivative Coefficients of a Rational Candidate
On all R, F has ascending rational coefficients [8/1, 21/2, -19/3, 11/2, 4/1]. Give its complete normalized ascending derivative coefficients as reduced fractions.
View answer and explanation
[21/2, -38/3, 33/2, 16/1]
Differentiate coefficientwise to obtain [21/2, -38/3, 33/2, 16/1]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-candidate-derivative · candidate
97. Derivative Coefficients of a Rational Candidate
On all R, F has ascending rational coefficients [9/1, 11/1, -20/3, 23/4, -21/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.
View answer and explanation
[11/1, -40/3, 69/4, -84/5]
Differentiate coefficientwise to obtain [11/1, -40/3, 69/4, -84/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-candidate-derivative · candidate
98. Derivative Coefficients of a Rational Candidate
On all R, F has ascending rational coefficients [10/1, 23/2, -7/1, 6/1, 22/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.
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[23/2, -14/1, 18/1, 88/5]
Differentiate coefficientwise to obtain [23/2, -14/1, 18/1, 88/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-candidate-derivative · candidate
99. Derivative Coefficients of a Rational Candidate
On all R, F has ascending rational coefficients [11/1, 12/1, -22/3, 25/4, -23/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.
View answer and explanation
[12/1, -44/3, 75/4, -92/5]
Differentiate coefficientwise to obtain [12/1, -44/3, 75/4, -92/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-candidate-derivative · candidate
100. Derivative Coefficients of a Rational Candidate
On all R, F has ascending rational coefficients [12/1, 25/2, -23/3, 13/2, 24/5]. Give its complete normalized ascending derivative coefficients as reduced fractions.
View answer and explanation
[25/2, -46/3, 39/2, 96/5]
Differentiate coefficientwise to obtain [25/2, -46/3, 39/2, 96/5]. The constant is removed, with [0/1] retained for a zero derivative; finite sample matching is not a coefficient-identity check.
Reviewed step-by-step derivation
Polynomial antiderivatives and the choice of constant · qf-density-primitive-candidate-derivative · candidate
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