integral of arctan x dx

xarctan x - 1/2ln (x^2 + 1) + C

integral of 6x × 2^x dx

(6x × 2^x)/(ln 2) - (6 × 2^x)/((ln 2)^2) + C

integral of 4xsin (3x) dx

-4/3xcos (3x) + 4/9sin (3x) + C

Integration by parts worksheet

The product rule run backwards. One factor is differentiated and the other integrated, and the trick is picking which way round: the one that gets simpler when differentiated is the one to differentiate.

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name Integration by Parts Date Period Evaluate each integral. Integrate by parts $\int 3 x^{5} \ln x d x$ $\int x^{2} \ln x d x$ $\int 5 x \ln x d x$ $\int 4 x^{4} \ln x d x$ $\int 2 \ln x d x$ Integrate logs and inverse trig by parts $\int \arctan x d x$ $\int - 4 \ln x d x$ $\int - 64 x^{3} \ln x d x$ $\int - 12 x \ln x d x$ $\int - 18 x^{2} \ln x d x$ Evaluate integrals by parts $\int_{1}^{e} x^{3} \ln x d x$ $\int_{0}^{\frac{π}{2}} x \sin x d x$ $\int e^{- 3 x} \cos x d x$ $\int_{0}^{2} x e^{x} d x$ $\int_{0}^{1} x e^{2 x} d x$ Choose $u$ and integrate by parts $\int \ln (x + 5) d x$ $\int 5 \ln (x + 6) d x$ $\int 6 x \cdot 2^{x} d x$ $\int \frac{\ln x}{x^{2}} d x$ $\int \frac{4 \ln x}{x^{3}} d x$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Integration by Parts

Date Period

Evaluate each integral.

Integrate by parts
$\int 3 x^{5} \ln x d x$
$\int x^{2} \ln x d x$
$\int 5 x \ln x d x$
$\int 4 x^{4} \ln x d x$
$\int 2 \ln x d x$

Integrate logs and inverse trig by parts
$\int \arctan x d x$
$\int - 4 \ln x d x$
$\int - 64 x^{3} \ln x d x$
$\int - 12 x \ln x d x$
$\int - 18 x^{2} \ln x d x$

Evaluate integrals by parts
$\int_{1}^{e} x^{3} \ln x d x$
$\int_{0}^{\frac{π}{2}} x \sin x d x$
$\int e^{- 3 x} \cos x d x$
$\int_{0}^{2} x e^{x} d x$
$\int_{0}^{1} x e^{2 x} d x$

Choose $u$ and integrate by parts
$\int \ln (x + 5) d x$
$\int 5 \ln (x + 6) d x$
$\int 6 x \cdot 2^{x} d x$
$\int \frac{\ln x}{x^{2}} d x$
$\int \frac{4 \ln x}{x^{3}} d x$



Sigma Prep · sigmaprep.io/worksheets Name Integration by PartsAnswer keyVersion 1 Date Period Evaluate each integral. Integrate by parts $\int 3 x^{5} \ln x d x$ $\frac{1}{2} x^{6} \ln x - \frac{1}{12} x^{6} + C$ $\int x^{2} \ln x d x$ $\frac{1}{3} x^{3} \ln x - \frac{1}{9} x^{3} + C$ $\int 5 x \ln x d x$ $\frac{5}{2} x^{2} \ln x - \frac{5}{4} x^{2} + C$ $\int 4 x^{4} \ln x d x$ $\frac{4}{5} x^{5} \ln x - \frac{4}{25} x^{5} + C$ $\int 2 \ln x d x$ $2 x \ln x - 2 x + C$ Integrate logs and inverse trig by parts $\int \arctan x d x$ $x \arctan x - \frac{1}{2} \ln ⁣ (x^{2} + 1) + C$ $\int - 4 \ln x d x$ $- 4 x \ln x + 4 x + C$ $\int - 64 x^{3} \ln x d x$ $- 16 x^{4} \ln x + 4 x^{4} + C$ $\int - 12 x \ln x d x$ $- 6 x^{2} \ln x + 3 x^{2} + C$ $\int - 18 x^{2} \ln x d x$ $- 6 x^{3} \ln x + 2 x^{3} + C$ Evaluate integrals by parts $\int_{1}^{e} x^{3} \ln x d x$ $\frac{3 e^{4} + 1}{16}$ $\int_{0}^{\frac{π}{2}} x \sin x d x$ $1$ $\int e^{- 3 x} \cos x d x$ $\frac{e^{- 3 x}}{10} (- 3 \cos x + \sin x) + C$ $\int_{0}^{2} x e^{x} d x$ $e^{2} + 1$ $\int_{0}^{1} x e^{2 x} d x$ $\frac{e^{2} + 1}{4}$ Choose $u$ and integrate by parts $\int \ln (x + 5) d x$ $(x + 5) \ln (x + 5) - x + C$ $\int 5 \ln (x + 6) d x$ $5 ((x + 6) \ln (x + 6) - x) + C$ $\int 6 x \cdot 2^{x} d x$ $\frac{6 x \cdot 2^{x}}{\ln 2} - \frac{6 \cdot 2^{x}}{{(\ln 2)}^{2}} + C$ $\int \frac{\ln x}{x^{2}} d x$ $- \frac{\ln x}{x} - \frac{1}{x} + C$ $\int \frac{4 \ln x}{x^{3}} d x$ $- \frac{2 \ln x}{x^{2}} - \frac{1}{x^{2}} + C$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Integration by PartsAnswer keyVersion 1

Date Period

Evaluate each integral.

Integrate by parts
$\int 3 x^{5} \ln x d x$ $\frac{1}{2} x^{6} \ln x - \frac{1}{12} x^{6} + C$
$\int x^{2} \ln x d x$ $\frac{1}{3} x^{3} \ln x - \frac{1}{9} x^{3} + C$
$\int 5 x \ln x d x$ $\frac{5}{2} x^{2} \ln x - \frac{5}{4} x^{2} + C$
$\int 4 x^{4} \ln x d x$ $\frac{4}{5} x^{5} \ln x - \frac{4}{25} x^{5} + C$
$\int 2 \ln x d x$ $2 x \ln x - 2 x + C$

Integrate logs and inverse trig by parts
$\int \arctan x d x$ $x \arctan x - \frac{1}{2} \ln ⁣ (x^{2} + 1) + C$
$\int - 4 \ln x d x$ $- 4 x \ln x + 4 x + C$
$\int - 64 x^{3} \ln x d x$ $- 16 x^{4} \ln x + 4 x^{4} + C$
$\int - 12 x \ln x d x$ $- 6 x^{2} \ln x + 3 x^{2} + C$
$\int - 18 x^{2} \ln x d x$ $- 6 x^{3} \ln x + 2 x^{3} + C$

Evaluate integrals by parts
$\int_{1}^{e} x^{3} \ln x d x$ $\frac{3 e^{4} + 1}{16}$
$\int_{0}^{\frac{π}{2}} x \sin x d x$ $1$
$\int e^{- 3 x} \cos x d x$ $\frac{e^{- 3 x}}{10} (- 3 \cos x + \sin x) + C$
$\int_{0}^{2} x e^{x} d x$ $e^{2} + 1$
$\int_{0}^{1} x e^{2 x} d x$ $\frac{e^{2} + 1}{4}$

Choose $u$ and integrate by parts
$\int \ln (x + 5) d x$ $(x + 5) \ln (x + 5) - x + C$
$\int 5 \ln (x + 6) d x$ $5 ((x + 6) \ln (x + 6) - x) + C$
$\int 6 x \cdot 2^{x} d x$ $\frac{6 x \cdot 2^{x}}{\ln 2} - \frac{6 \cdot 2^{x}}{{(\ln 2)}^{2}} + C$
$\int \frac{\ln x}{x^{2}} d x$ $- \frac{\ln x}{x} - \frac{1}{x} + C$
$\int \frac{4 \ln x}{x^{3}} d x$ $- \frac{2 \ln x}{x^{2}} - \frac{1}{x^{2}} + C$

How to do these

$\int x e^{2 x} d x$

Differentiate the x and integrate the exponential.
That gives x times e to the 2x over 2, minus the integral of e to the 2x over 2.
The integral left is easy, so the answer is x e to the 2x over 2, minus e to the 2x over 4.

The answer is $\frac{x e^{2 x}}{2} - \frac{e^{2 x}}{4} + C$ .

Where students go wrong

Differentiating the wrong factor. Differentiate the exponential and integrate the x and the next integral is worse than the one you started with. The polynomial is what gets simpler, so that is the one to differentiate, unless there is a log, which has to be.

Also called integrating by parts, u dv, LIATE or reverse product rule.

What you can put on this worksheet

Integration by parts: $\int 4 x \sin (3 x) d x$ → $- \frac{4}{3} x \cos (3 x) + \frac{4}{9} \sin (3 x) + C$
ln x and inverse trig: $\int 12 x \ln x d x$ → $6 x^{2} \ln x - 3 x^{2} + C$
Cyclic and definite: $\int_{0}^{π} 3 x \cos x d x$ → $- 6$
Other bases, roots times logs, x^2 over e^kx, ln of x + a: $\int 5 x \ln x d x$ → $\frac{5}{2} x^{2} \ln x - \frac{5}{4} x^{2} + C$

Questions about these worksheets

Yes. Every integration by parts 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 parts 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 4: integration by parts, ln x and inverse trig, cyclic and definite and other bases, roots times logs, x^2 over e^kx, ln of x + a. Tick as many as you want and set how many of each, or let it spread them evenly.

Integration by parts worksheet

Select your difficulty

Integrate by parts

Integrate logs and inverse trig by parts

Evaluate integrals by parts

Choose uuu and integrate by parts

Integrate by parts

Integrate logs and inverse trig by parts

Evaluate integrals by parts

Choose uuu and integrate by parts

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these integration by parts worksheets free?+

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

Are these integration by parts worksheets printable?+

What grade are these integration by parts worksheets for?+

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

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

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

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

What do students get wrong with integration by parts?+

Choose $u$ and integrate by parts

Choose $u$ and integrate by parts