Polynomial Integral Worksheet Generator
Each problem is the indefinite integral of a polynomial in x. The answer key applies the reverse power rule term by term, leaves a +C, and reports any fractional coefficients in lowest terms with parentheses for readability.
Practice where the fractions are the point
The reverse power rule is short to state and awkward to execute. Raising the exponent is easy; dividing by the new exponent is where the fractions arrive, and fractions are what makes a page of otherwise routine integrals slow and error-prone.
This generator produces sheets aimed squarely at that difficulty. Each problem is the indefinite integral of a polynomial with integer coefficients, and each answer is given in fully reduced form with its constant of integration.
A polynomial integrand keeps the technique fixed so the arithmetic can be the challenge. Dividing 3 by 2 to get three halves, or 9 by 3 to get a clean 3, is the step that separates a confident answer from a hesitant one, and only repetition makes it automatic.
The answer key reduces every coefficient to lowest terms and brackets any that remains fractional, so an answer of six over four is marked against three halves rather than against an unreduced equivalent. Matching the key exactly therefore means having finished the arithmetic, not merely started it.
How to use this calculator
- Set how many problems you want Between 1 and 30. A larger request is clamped to 30, and an entry that is not a number falls back to ten.
- Pick the difficulty Easy gives linear integrands, medium quadratic and hard cubic. Each step up adds one term to integrate and one more coefficient to divide.
- Set a seed, or leave it blank A whole number reproduces the same sheet exactly. Blank picks one at random and prints it with the worksheet, so any sheet can be recreated afterwards.
- Integrate term by term, then check the key The problems come first and the full answer key last, so the sheet can be worked before the answers are in view.
How the problems and answers are built
The seed is read first: a whole number is used as given, and a blank or unreadable entry is replaced by a random one that is then reported alongside the sheet. It initialises a mulberry32 generator whose output depends only on that starting value, which is what makes a seeded worksheet reproducible anywhere.
The difficulty sets the degree of the integrand — one, two or three — and the coefficient range, which runs to six, seven and nine respectively. For each problem the generator draws an integer coefficient for every power up to that degree, redrawing the leading one from a smaller non-zero range if it comes out as zero, so an easy integrand is genuinely linear and a hard one genuinely cubic.
The answer is built from the coefficient list rather than by manipulating the printed problem. Each coefficient becomes a numerator and the exponent one higher becomes its denominator, giving the reverse power rule directly in fractional form. Nothing is evaluated as a decimal at any point, which is why the key shows exact fractions rather than rounded values.
Each fraction is then reduced to lowest terms, and one that does not reduce to a whole number is printed inside brackets so a coefficient like three halves reads unambiguously against the power that follows it. The constant of integration is appended to every answer.
What each input means
- n Number of problems — form field “Number of problems”
- How many integrals appear on the sheet, clamped between 1 and 30. Units: problems.
- d Difficulty — form field “Difficulty”
- Easy, medium or hard, setting the integrand's degree to 1, 2 or 3 and the coefficient range to ±6, ±7 or ±9.
- s Seed — form field “Seed (optional)”
- Optional whole number fixing the entire sheet. Reported with the output whether you supplied it or not.
Worked examples
Every number below is produced by the same calculation engine the tool above runs. Nothing here is typed by hand, so the walkthrough cannot drift from what you get when you enter the same values yourself.
Ten linear integrands
Ten easy problems from the seed 20260820. Each integrand is a linear polynomial, so each answer is a quadratic plus a constant.
Inputs Number of problems = 10, Difficulty = easy, Seed (optional) = 20260820
- Seed 20260820 (use this seed to regenerate the same worksheet)
- Count 10 problems
- Problem 1. Integrate: ∫ (6x − 1) dx
- Problem 2. Integrate: ∫ (5x + 1) dx
- Problem 3. Integrate: ∫ (3x − 5) dx
- Problem 4. Integrate: ∫ (−5x − 5) dx
- Problem 5. Integrate: ∫ (−x − 1) dx
- Problem 6. Integrate: ∫ (3x − 5) dx
- Problem 7. Integrate: ∫ (−3x − 2) dx
- Problem 8. Integrate: ∫ (4x + 5) dx
- Problem 9. Integrate: ∫ (−6x) dx
- Problem 10. Integrate: ∫ (3x + 2) dx
- Answer key 1. 3x^2 − x + C; 2. (5/2)x^2 + x + C; 3. (3/2)x^2 − 5x + C; 4. −(5/2)x^2 − 5x + C; 5. −(1/2)x^2 − x + C; 6. (3/2)x^2 − 5x + C; 7. −(3/2)x^2 − 2x + C; 8. 2x^2 + 5x + C; 9. −3x^2 + C; 10. (3/2)x^2 + 2x + C
Result 10-problem worksheet — Polynomial integrals (easy) (seed 20260820)
The x term always produces a fraction with denominator 2, since its exponent rises from one to two. Roughly half the problems therefore carry a bracketed coefficient, which is exactly the arithmetic the easy level exists to drill.
The constant term integrates to a plain multiple of x, with no division at all. Having both kinds of term in every problem keeps the contrast between them visible.
The same seed at hard difficulty
Ten problems from the same seed, but with cubic integrands. This isolates what the difficulty setting changes.
Inputs Number of problems = 10, Difficulty = hard, Seed (optional) = 20260820
- Seed 20260820 (use this seed to regenerate the same worksheet)
- Count 10 problems
- Problem 1. Integrate: ∫ (8x^3 + x^2 + 8x − 1) dx
- Problem 2. Integrate: ∫ (−8x^3 − 8x^2 + 4x − 8) dx
- Problem 3. Integrate: ∫ (−8x^3 − 3x^2 + x − 1) dx
- Problem 4. Integrate: ∫ (−8x^3 − 3x + 4) dx
- Problem 5. Integrate: ∫ (−9x^3 + 6x + 7) dx
- Problem 6. Integrate: ∫ (7x^3 + 5x + 3) dx
- Problem 7. Integrate: ∫ (−4x^3 + 9x^2 + 9x − 5) dx
- Problem 8. Integrate: ∫ (−2x^3 + 6x + 5) dx
- Problem 9. Integrate: ∫ (3x^3 + 6x^2 − 4x − 9) dx
- Problem 10. Integrate: ∫ (−8x^3 − 2x^2 + 6x − 5) dx
- Answer key 1. 2x^4 + (1/3)x^3 + 4x^2 − x + C; 2. −2x^4 − (8/3)x^3 + 2x^2 − 8x + C; 3. −2x^4 − x^3 + (1/2)x^2 − x + C; 4. −2x^4 − (3/2)x^2 + 4x + C; 5. −(9/4)x^4 + 3x^2 + 7x + C; 6. (7/4)x^4 + (5/2)x^2 + 3x + C; 7. −x^4 + 3x^3 + (9/2)x^2 − 5x + C; 8. −(1/2)x^4 + 3x^2 + 5x + C; 9. (3/4)x^4 + 2x^3 − 2x^2 − 9x + C; 10. −2x^4 − (2/3)x^3 + 3x^2 − 5x + C
Result 10-problem worksheet — Polynomial integrals (hard) (seed 20260820)
The answers are now quartics, with four terms each and denominators running from 1 up to 4. A denominator of 4 rarely cancels, so the hard sheets carry noticeably more bracketed coefficients than the easy ones.
The numbers differ entirely from the easy sheet despite the identical seed. Four coefficients are drawn per problem instead of two, so the sequence of draws diverges immediately — a seed reproduces a sheet only when the count and difficulty match too.
Reading the result
Reading a bracketed coefficient
Brackets mark a coefficient that stayed fractional after reduction, and they are there to keep it from running into the power beside it. A coefficient printed without them is a whole number, which means the division came out exactly.
Why every answer ends in + C
Indefinite integration recovers an antiderivative only up to a constant, because differentiating any constant gives zero. The + C stands for that whole family, and an answer without it is incomplete rather than merely untidy.
Checking your own answer
Differentiate it. If the reverse power rule was applied correctly the derivative returns the integrand exactly, which is a check that needs no reference to the key at all.
When you would use this
Drilling until the fractions stop slowing you down
Working several sheets at one difficulty builds the fluency that makes the division automatic, which is what allows attention to move on to substitution and by parts later.
Setting reusable homework
Recording the seed, count and difficulty lets a sheet be recreated exactly, months later, with nothing stored. Different seeds give different students different numbers on the same task.
Assumptions and limitations
What this calculator assumes
- Integrands are polynomials with integer coefficients and non-negative whole exponents.
- The leading coefficient is never zero, so each integrand has the degree its difficulty implies.
- Answer coefficients are exact fractions in lowest terms, never decimals.
- The same seed, count and difficulty reproduce the same sheet exactly.
Where it stops being the right tool
- Indefinite integrals only. For definite ones, the dedicated definite-integral solver is the right tool.
- Polynomials only: no trigonometric, exponential or logarithmic integrands, and therefore no substitution or by-parts practice.
- No negative or fractional exponents, so the logarithmic case that the power rule cannot handle never appears.
- One difficulty per sheet, and at most 30 problems.
Common mistakes
Raising the exponent but forgetting to divide
Why it happens. The two halves of the rule are not equally memorable. Raising the power is the visible change; dividing by the new power is the step that quietly disappears under time pressure.
How to avoid it. Differentiate one term of your answer. If it comes back as a multiple of the original term rather than the term itself, the division is what is missing.
Dividing by the old exponent
Why it happens. Both numbers are in front of you, and the rule reads as a single motion, so it is easy to divide by the exponent that was there rather than the one just created.
How to avoid it. Do it in two steps. Write the raised power first, then divide by exactly what you have written — never by what you started with.
Leaving a coefficient unreduced
Why it happens. Six over four is a correct answer to the division, and stopping there feels like finishing. The key reduces, so a correct-but-unreduced answer will not match it.
How to avoid it. Reduce every fraction before comparing. Where the key shows no brackets, the division produced a whole number and yours should too.
Frequently asked questions
Why is there a +C in every answer?
Because indefinite integration recovers an antiderivative only up to an additive constant: any constant differentiates to zero, so infinitely many functions share the same derivative. The + C records that the original could differ from the printed answer by any constant.
Can the coefficients be fractions?
Yes, and that is largely the point. Dividing by the raised exponent turns a coefficient of 3 on an x term into three halves. The key keeps such fractions in lowest terms and brackets them so they read clearly against the power.
Are there definite integrals?
No. Every problem here is indefinite, so every answer is an expression with + C rather than a number. For definite integrals with limits, use the dedicated solver in the calculus category.
What does the difficulty setting change?
The degree of the integrand and the size of its coefficients. Easy gives linear integrands with coefficients up to six in magnitude, medium quadratic ones up to seven, and hard cubic ones up to nine. Each step adds a term to integrate.