Definite integrals worksheet

Find an antiderivative, put the top limit in, put the bottom limit in, subtract. That is the fundamental theorem, and it is why a question about area can be answered by algebra.

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Pick more than one for a sheet that mixes them.



Sigma Prep · sigmaprep.io/worksheets Name Definite Integrals Date Period Evaluate each integral. $\int_{0}^{2} 4 - 48 x^{3} d x$ $\int_{0}^{1} - 12 x^{2} {(2 x^{3} + 2)}^{2} d x$ $\int_{- 3}^{- 2} 8 x d x$ $\int_{0}^{2} 32 x {(4 x^{2} + 3)}^{2} d x$ $\int_{1}^{2} - 36 x^{2} d x$ $\int_{0}^{1} 18 x^{2} {(2 x^{3} + 4)}^{2} d x$ $\int_{1}^{2} - 12 x d x$ $\int_{0}^{3} 18 x^{2} {(2 x^{3} + 3)}^{2} d x$ $\int_{1}^{4} 16 x^{3} + 2 d x$ $\int_{0}^{1} 3 x^{2} {(x^{3} + 7)}^{2} d x$ $\int_{0}^{3} - 2 d x$ $\int_{0}^{1} - 48 x^{2} {(4 x^{3} + 8)}^{2} d x$ $\int_{- 2}^{- 1} 3 d x$ $\int_{0}^{1} 12 x {(2 x^{2} + 7)}^{2} d x$ $\int_{1}^{5} - 64 x^{3} d x$ $\int_{0}^{3} 2 x {(x^{2} + 9)}^{2} d x$ $\int_{- 2}^{- 1} 64 x^{3} d x$ $\int_{0}^{1} 24 x {(3 x^{2} + 3)}^{2} d x$ $\int_{1}^{5} - 18 x^{2} - 8 x d x$ $\int_{0}^{2} - 2 x {(x^{2} + 1)}^{2} d x$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Definite Integrals

Date Period

Evaluate each integral.

$\int_{0}^{2} 4 - 48 x^{3} d x$
$\int_{0}^{1} - 12 x^{2} {(2 x^{3} + 2)}^{2} d x$
$\int_{- 3}^{- 2} 8 x d x$
$\int_{0}^{2} 32 x {(4 x^{2} + 3)}^{2} d x$
$\int_{1}^{2} - 36 x^{2} d x$
$\int_{0}^{1} 18 x^{2} {(2 x^{3} + 4)}^{2} d x$
$\int_{1}^{2} - 12 x d x$
$\int_{0}^{3} 18 x^{2} {(2 x^{3} + 3)}^{2} d x$
$\int_{1}^{4} 16 x^{3} + 2 d x$
$\int_{0}^{1} 3 x^{2} {(x^{3} + 7)}^{2} d x$
$\int_{0}^{3} - 2 d x$
$\int_{0}^{1} - 48 x^{2} {(4 x^{3} + 8)}^{2} d x$
$\int_{- 2}^{- 1} 3 d x$
$\int_{0}^{1} 12 x {(2 x^{2} + 7)}^{2} d x$
$\int_{1}^{5} - 64 x^{3} d x$
$\int_{0}^{3} 2 x {(x^{2} + 9)}^{2} d x$
$\int_{- 2}^{- 1} 64 x^{3} d x$
$\int_{0}^{1} 24 x {(3 x^{2} + 3)}^{2} d x$
$\int_{1}^{5} - 18 x^{2} - 8 x d x$
$\int_{0}^{2} - 2 x {(x^{2} + 1)}^{2} d x$



Sigma Prep · sigmaprep.io/worksheets Name Definite IntegralsAnswer keyVersion 1 Date Period Evaluate each integral. $\int_{0}^{2} 4 - 48 x^{3} d x$ $- 184$ $\int_{0}^{1} - 12 x^{2} {(2 x^{3} + 2)}^{2} d x$ $- \frac{112}{3}$ $\int_{- 3}^{- 2} 8 x d x$ $- 20$ $\int_{0}^{2} 32 x {(4 x^{2} + 3)}^{2} d x$ $\frac{27328}{3}$ $\int_{1}^{2} - 36 x^{2} d x$ $- 84$ $\int_{0}^{1} 18 x^{2} {(2 x^{3} + 4)}^{2} d x$ $152$ $\int_{1}^{2} - 12 x d x$ $- 18$ $\int_{0}^{3} 18 x^{2} {(2 x^{3} + 3)}^{2} d x$ $185166$ $\int_{1}^{4} 16 x^{3} + 2 d x$ $1026$ $\int_{0}^{1} 3 x^{2} {(x^{3} + 7)}^{2} d x$ $\frac{169}{3}$ $\int_{0}^{3} - 2 d x$ $- 6$ $\int_{0}^{1} - 48 x^{2} {(4 x^{3} + 8)}^{2} d x$ $- \frac{4864}{3}$ $\int_{- 2}^{- 1} 3 d x$ $3$ $\int_{0}^{1} 12 x {(2 x^{2} + 7)}^{2} d x$ $386$ $\int_{1}^{5} - 64 x^{3} d x$ $- 9984$ $\int_{0}^{3} 2 x {(x^{2} + 9)}^{2} d x$ $1701$ $\int_{- 2}^{- 1} 64 x^{3} d x$ $- 240$ $\int_{0}^{1} 24 x {(3 x^{2} + 3)}^{2} d x$ $252$ $\int_{1}^{5} - 18 x^{2} - 8 x d x$ $- 840$ $\int_{0}^{2} - 2 x {(x^{2} + 1)}^{2} d x$ $- \frac{124}{3}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Definite IntegralsAnswer keyVersion 1

Date Period

Evaluate each integral.

$\int_{0}^{2} 4 - 48 x^{3} d x$ $- 184$
$\int_{0}^{1} - 12 x^{2} {(2 x^{3} + 2)}^{2} d x$ $- \frac{112}{3}$
$\int_{- 3}^{- 2} 8 x d x$ $- 20$
$\int_{0}^{2} 32 x {(4 x^{2} + 3)}^{2} d x$ $\frac{27328}{3}$
$\int_{1}^{2} - 36 x^{2} d x$ $- 84$
$\int_{0}^{1} 18 x^{2} {(2 x^{3} + 4)}^{2} d x$ $152$
$\int_{1}^{2} - 12 x d x$ $- 18$
$\int_{0}^{3} 18 x^{2} {(2 x^{3} + 3)}^{2} d x$ $185166$
$\int_{1}^{4} 16 x^{3} + 2 d x$ $1026$
$\int_{0}^{1} 3 x^{2} {(x^{3} + 7)}^{2} d x$ $\frac{169}{3}$
$\int_{0}^{3} - 2 d x$ $- 6$
$\int_{0}^{1} - 48 x^{2} {(4 x^{3} + 8)}^{2} d x$ $- \frac{4864}{3}$
$\int_{- 2}^{- 1} 3 d x$ $3$
$\int_{0}^{1} 12 x {(2 x^{2} + 7)}^{2} d x$ $386$
$\int_{1}^{5} - 64 x^{3} d x$ $- 9984$
$\int_{0}^{3} 2 x {(x^{2} + 9)}^{2} d x$ $1701$
$\int_{- 2}^{- 1} 64 x^{3} d x$ $- 240$
$\int_{0}^{1} 24 x {(3 x^{2} + 3)}^{2} d x$ $252$
$\int_{1}^{5} - 18 x^{2} - 8 x d x$ $- 840$
$\int_{0}^{2} - 2 x {(x^{2} + 1)}^{2} d x$ $- \frac{124}{3}$

How to do these

$\int_{0}^{2} 3 x^{2} d x$

The antiderivative is x cubed.
At 2 that is 8, at 0 it is 0.
Subtract.

The answer is $8$ .

Where students go wrong

Subtracting the wrong way round. It is top minus bottom, and swapping them flips the sign of every answer on the page.

Also called fundamental theorem of calculus, area under a curve or evaluating integrals.

What you can put on this worksheet

Definite integrals: $\int_{1}^{3} 4 x + 2 d x$ → $20$
The area under a curve: Area under $y = - 2 x^{2} + 2 x + 4$ from $x = - 1$ to $x = 2$ → $9$
Substitution with changed bounds: $\int_{0}^{2} - 24 x (3 x^{2} + 1) d x$ → $- 336$

Questions about these worksheets

Yes. Every definite integrals sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the definite integrals questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are two levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 3: definite integrals, the area under a curve and substitution with changed bounds. Tick as many as you want and set how many of each, or let it spread them evenly.

Subtracting the wrong way round. It is top minus bottom, and swapping them flips the sign of every answer on the page.

Definite integrals worksheet

Select your difficulty

Definite Integrals

Definite IntegralsAnswer keyVersion 1

How to do these

Where students go wrong

What you can put on this worksheet

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