Practice Integration by Parts in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

An integration technique based on the product rule: ∫u dv=uv−∫v du\int u\,dv = uv - \int v\,du. Used when the integrand is a product of two functions.

The product rule for derivatives says (uv)′=u′v+uv′(uv)' = u'v + uv'. Rearranging and integrating gives integration by parts. The idea is to trade your original integral for a (hopefully easier) one. You're transferring the derivative from one factor to the other.

Showing a random 20 of 50 problems.

Example 1

medium
Evaluate ∫xsec⁡2x dx\int x\sec^2 x\,dx.

Example 2

hard
Evaluate ∫x3ex dx\int x^3 e^x\,dx.

Example 3

medium
Evaluate ∫01arctan⁡x dx\int_0^1 \arctan x\,dx.

Example 4

hard
Evaluate ∫1ex2ln⁡x dx\int_1^{e} x^2 \ln x\,dx.

Example 5

medium
Evaluate ∫arctan⁡x dx\int \arctan x\,dx.

Example 6

hard
Evaluate ∫(ln⁡x)2 dx\int (\ln x)^2\,dx.

Example 7

medium
Evaluate ∫0π/2xsin⁡x dx\int_0^{\pi/2} x\sin x\,dx.

Example 8

easy
State the integration by parts formula.

Example 9

hard
Derive a reduction formula for In=∫(ln⁡x)n dxI_n = \int (\ln x)^n\,dx.

Example 10

medium
Evaluate ∫x2ex dx\int x^2 e^x\,dx.

Example 11

easy
Evaluate ∫3xcos⁡x dx\int 3x\cos x\,dx.

Example 12

hard
Find ∫exsin⁡x dx\displaystyle\int e^x \sin x\,dx.

Example 13

challenge
Evaluate ∫excos⁡x dx\int e^x\cos x\,dx (the cyclic case).

Example 14

medium
Evaluate ∫x2ex dx\displaystyle\int x^2 e^x \, dx using integration by parts twice.

Example 15

easy
Find ∫xcos⁡x dx\displaystyle\int x\cos x\,dx.

Example 16

medium
Evaluate ∫01xex dx\int_0^1 x e^x\,dx.

Example 17

medium
Evaluate ∫xcos⁡x dx\int x\cos x\,dx.

Example 18

hard
Evaluate ∫01x2e−x dx\int_0^1 x^2 e^{-x}\,dx.

Example 19

medium
Evaluate ∫ln⁡(x2) dx\int \ln(x^2)\,dx.

Example 20

challenge
Use IBP to show ∫0∞xe−x dx=1\int_0^{\infty} x e^{-x}\,dx = 1.
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Related Concepts

IntegralDerivative