Integration by substitution worksheet

The chain rule leaves the derivative of the inside sitting as a factor, so seeing it there is the signal. Call the inside u, and what is left is a power rule in u.



Sigma Prep · sigmaprep.io/worksheets Name Integration by Substitution Date Period Evaluate each integral. $\int 30 x {(x^{2} + 8)}^{4} d x$ $\int \frac{6 x}{- x^{2} - 3} d x$ $\int - 24 {(- 2 x + 3)}^{3} d x$ $\int - 24 x e^{- 3 x^{2}} d x$ $\int - 48 x {(- 4 x^{2} - 7)}^{5} d x$ $\int \frac{4}{- 4 x + 2} d x$ $\int 120 x {(5 x^{2} - 8)}^{3} d x$ $\int 60 x^{2} e^{4 x^{3}} d x$ $\int - 36 x {(- x^{2} + 3)}^{5} d x$ $\int \frac{30 x}{- 5 x^{2} - 4} d x$ $\int - 48 {(- 4 x + 5)}^{5} d x$ $\int \frac{1}{- x + 5} d x$ $\int - 30 x {(- x^{2} - 6)}^{4} d x$ $\int \frac{- 10 x}{5 x^{2} + 1} d x$ $\int 48 {(2 x + 7)}^{5} d x$ $\int 6 x e^{- x^{2}} d x$ $\int - 50 x {(- 5 x^{2} - 8)}^{4} d x$ $\int \frac{5}{- 5 x + 2} d x$ $\int - 64 {(- 4 x - 2)}^{3} d x$ $\int - 12 x e^{2 x^{2}} d x$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Integration by Substitution

Date Period

Evaluate each integral.

$\int 30 x {(x^{2} + 8)}^{4} d x$
$\int \frac{6 x}{- x^{2} - 3} d x$
$\int - 24 {(- 2 x + 3)}^{3} d x$
$\int - 24 x e^{- 3 x^{2}} d x$
$\int - 48 x {(- 4 x^{2} - 7)}^{5} d x$
$\int \frac{4}{- 4 x + 2} d x$
$\int 120 x {(5 x^{2} - 8)}^{3} d x$
$\int 60 x^{2} e^{4 x^{3}} d x$
$\int - 36 x {(- x^{2} + 3)}^{5} d x$
$\int \frac{30 x}{- 5 x^{2} - 4} d x$
$\int - 48 {(- 4 x + 5)}^{5} d x$
$\int \frac{1}{- x + 5} d x$
$\int - 30 x {(- x^{2} - 6)}^{4} d x$
$\int \frac{- 10 x}{5 x^{2} + 1} d x$
$\int 48 {(2 x + 7)}^{5} d x$
$\int 6 x e^{- x^{2}} d x$
$\int - 50 x {(- 5 x^{2} - 8)}^{4} d x$
$\int \frac{5}{- 5 x + 2} d x$
$\int - 64 {(- 4 x - 2)}^{3} d x$
$\int - 12 x e^{2 x^{2}} d x$



Sigma Prep · sigmaprep.io/worksheets Name Integration by SubstitutionAnswer keyVersion 1 Date Period Evaluate each integral. $\int 30 x {(x^{2} + 8)}^{4} d x$ $3 {(x^{2} + 8)}^{5} + C$ $\int \frac{6 x}{- x^{2} - 3} d x$ $- 3 \ln ∣ - x^{2} - 3 ∣ + C$ $\int - 24 {(- 2 x + 3)}^{3} d x$ $3 {(- 2 x + 3)}^{4} + C$ $\int - 24 x e^{- 3 x^{2}} d x$ $4 e^{- 3 x^{2}} + C$ $\int - 48 x {(- 4 x^{2} - 7)}^{5} d x$ ${(- 4 x^{2} - 7)}^{6} + C$ $\int \frac{4}{- 4 x + 2} d x$ $- \ln ∣ - 4 x + 2 ∣ + C$ $\int 120 x {(5 x^{2} - 8)}^{3} d x$ $3 {(5 x^{2} - 8)}^{4} + C$ $\int 60 x^{2} e^{4 x^{3}} d x$ $5 e^{4 x^{3}} + C$ $\int - 36 x {(- x^{2} + 3)}^{5} d x$ $3 {(- x^{2} + 3)}^{6} + C$ $\int \frac{30 x}{- 5 x^{2} - 4} d x$ $- 3 \ln ∣ - 5 x^{2} - 4 ∣ + C$ $\int - 48 {(- 4 x + 5)}^{5} d x$ $2 {(- 4 x + 5)}^{6} + C$ $\int \frac{1}{- x + 5} d x$ $- \ln ∣ - x + 5 ∣ + C$ $\int - 30 x {(- x^{2} - 6)}^{4} d x$ $3 {(- x^{2} - 6)}^{5} + C$ $\int \frac{- 10 x}{5 x^{2} + 1} d x$ $- \ln ∣ 5 x^{2} + 1 ∣ + C$ $\int 48 {(2 x + 7)}^{5} d x$ $4 {(2 x + 7)}^{6} + C$ $\int 6 x e^{- x^{2}} d x$ $- 3 e^{- x^{2}} + C$ $\int - 50 x {(- 5 x^{2} - 8)}^{4} d x$ ${(- 5 x^{2} - 8)}^{5} + C$ $\int \frac{5}{- 5 x + 2} d x$ $- \ln ∣ - 5 x + 2 ∣ + C$ $\int - 64 {(- 4 x - 2)}^{3} d x$ $4 {(- 4 x - 2)}^{4} + C$ $\int - 12 x e^{2 x^{2}} d x$ $- 3 e^{2 x^{2}} + C$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Integration by SubstitutionAnswer keyVersion 1

Date Period

Evaluate each integral.

$\int 30 x {(x^{2} + 8)}^{4} d x$ $3 {(x^{2} + 8)}^{5} + C$
$\int \frac{6 x}{- x^{2} - 3} d x$ $- 3 \ln ∣ - x^{2} - 3 ∣ + C$
$\int - 24 {(- 2 x + 3)}^{3} d x$ $3 {(- 2 x + 3)}^{4} + C$
$\int - 24 x e^{- 3 x^{2}} d x$ $4 e^{- 3 x^{2}} + C$
$\int - 48 x {(- 4 x^{2} - 7)}^{5} d x$ ${(- 4 x^{2} - 7)}^{6} + C$
$\int \frac{4}{- 4 x + 2} d x$ $- \ln ∣ - 4 x + 2 ∣ + C$
$\int 120 x {(5 x^{2} - 8)}^{3} d x$ $3 {(5 x^{2} - 8)}^{4} + C$
$\int 60 x^{2} e^{4 x^{3}} d x$ $5 e^{4 x^{3}} + C$
$\int - 36 x {(- x^{2} + 3)}^{5} d x$ $3 {(- x^{2} + 3)}^{6} + C$
$\int \frac{30 x}{- 5 x^{2} - 4} d x$ $- 3 \ln ∣ - 5 x^{2} - 4 ∣ + C$
$\int - 48 {(- 4 x + 5)}^{5} d x$ $2 {(- 4 x + 5)}^{6} + C$
$\int \frac{1}{- x + 5} d x$ $- \ln ∣ - x + 5 ∣ + C$
$\int - 30 x {(- x^{2} - 6)}^{4} d x$ $3 {(- x^{2} - 6)}^{5} + C$
$\int \frac{- 10 x}{5 x^{2} + 1} d x$ $- \ln ∣ 5 x^{2} + 1 ∣ + C$
$\int 48 {(2 x + 7)}^{5} d x$ $4 {(2 x + 7)}^{6} + C$
$\int 6 x e^{- x^{2}} d x$ $- 3 e^{- x^{2}} + C$
$\int - 50 x {(- 5 x^{2} - 8)}^{4} d x$ ${(- 5 x^{2} - 8)}^{5} + C$
$\int \frac{5}{- 5 x + 2} d x$ $- \ln ∣ - 5 x + 2 ∣ + C$
$\int - 64 {(- 4 x - 2)}^{3} d x$ $4 {(- 4 x - 2)}^{4} + C$
$\int - 12 x e^{2 x^{2}} d x$ $- 3 e^{2 x^{2}} + C$

How to do these

$\int 6 x (x^{2} + 1)^{2} d x$

Let u be x squared plus 1.
Then du is 2x dx, and 6x dx is 3 du.
Integrate 3u squared.

The answer is $(x^{2} + 1)^{3} + C$ .

Where students go wrong

Substituting for u without dealing with dx. Every x in the integral has to go, the dx included, and swapping only the parenthesis leaves an expression in two variables.

Also called u substitution, change of variable or reverse chain rule.

What you can put on this worksheet

Integration by substitution: $\int 24 {(2 x + 1)}^{2} d x$ → $4 {(2 x + 1)}^{3} + C$
Substitution giving a log or an exponential: $\int \frac{- 9}{- 3 x + 3} d x$ → $3 \ln ∣ - 3 x + 3 ∣ + C$

Questions about these worksheets

Yes. Every integration by substitution 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 integration by substitution 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 three 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 2: integration by substitution and substitution giving a log or an exponential. Tick as many as you want and set how many of each, or let it spread them evenly.

Substituting for u without dealing with dx. Every x in the integral has to go, the dx included, and swapping only the parenthesis leaves an expression in two variables.

Integration by substitution worksheet

Integration by Substitution

Integration by SubstitutionAnswer keyVersion 1

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these integration by substitution worksheets free?+

Do the integration by substitution worksheets come with an answer key?+

Are these integration by substitution worksheets printable?+

What grade are these integration by substitution worksheets for?+

Can I get a different version of the same integration by substitution worksheet?+

Is this a free alternative to Kuta Software for integration by substitution?+

Can I choose how hard the integration by substitution questions are?+

What kinds of question are on a integration by substitution worksheet?+

What do students get wrong with integration by substitution?+