Integration by Parts Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Integration by Parts.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

An integration technique based on the product rule: ∫u dv=uvβˆ’βˆ«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)β€²=uβ€²v+uvβ€²(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.

Read the full concept explanation β†’

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Integration by parts uses ∫u dv=uvβˆ’βˆ«v du\int u\,dv=uv-\int v\,du to transfer the derivative from one factor to the other.

Common stuck point: The procedure for integration by parts is the easy part; the trap is picking uu and dvdv backward so the new integral is harder. Asking "Is the integrand a product of unlike functions where differentiating one factor simplifies it, with no inner-derivative match for substitution?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the integrand a product of unlike functions where differentiating one factor simplifies it, with no inner-derivative match for substitution?

Worked Examples

Example 1

easy
Find ∫xex dx\displaystyle\int x e^x\,dx.

Answer

ex(xβˆ’1)+Ce^x(x-1) + C

First step

1
LIATE: u=xu = x, dv=ex dxdv = e^x\,dx; then du=dxdu = dx, v=exv = e^x.

Full solution

  1. 2
    ∫xex dx=xexβˆ’βˆ«ex dx=xexβˆ’ex+C\int xe^x\,dx = xe^x - \int e^x\,dx = xe^x - e^x + C.
  2. 3
    Factor: ex(xβˆ’1)+Ce^x(x-1) + C.
LIATE places algebraic before exponential, so u=xu = x. One IBP step reduces the remaining integral to something immediate.

Example 2

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

Example 3

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

Example 4

medium
Evaluate ∫0Ο€/2xsin⁑x dx\int_0^{\pi/2} x\sin x\,dx.

Example 5

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

Example 6

challenge
Prove the reduction formula ∫xneax dx=xneaxaβˆ’na∫xnβˆ’1eax dx\int x^n e^{ax}\,dx = \frac{x^n e^{ax}}{a} - \frac{n}{a}\int x^{n-1} e^{ax}\,dx for aβ‰ 0a \neq 0.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

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

Example 2

medium
Find ∫ln⁑x dx\displaystyle\int \ln x\,dx.

Example 3

easy
Evaluate ∫xex dx\int x e^x\,dx.

Example 4

easy
In ∫xcos⁑x dx\int x\cos x\,dx, what should uu be by LIATE?

Example 5

easy
Evaluate ∫ln⁑x dx\int \ln x\,dx.

Example 6

easy
State the integration by parts formula.

Example 7

easy
In ∫xex dx\int x e^x\,dx, identify vv if dv=ex dxdv=e^x\,dx.

Example 8

easy
Evaluate ∫xsin⁑x dx\int x\sin x\,dx.

Example 9

easy
In ∫xln⁑x dx\int x\ln x\,dx, what is uu by LIATE?

Example 10

easy
Why does integration by parts work? (one sentence)

Example 11

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

Example 12

medium
Evaluate ∫xln⁑x dx\int x\ln x\,dx.

Example 13

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

Example 14

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

Example 15

medium
Evaluate ∫x2ln⁑x dx\int x^2\ln x\,dx.

Example 16

medium
Evaluate ∫x2sin⁑x dx\int x^2\sin x\,dx.

Example 17

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

Example 18

challenge
Evaluate ∫1eln⁑x dx\int_1^e \ln x\,dx using parts.

Example 19

challenge
Derive a reduction-style result: show ∫xnex dx=xnexβˆ’n∫xnβˆ’1ex dx\int x^n e^x\,dx=x^n e^x-n\int x^{n-1} e^x\,dx.

Example 20

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

Example 21

medium
Evaluate ∫(2x+1)ex dx\int (2x+1)e^x\,dx.

Example 22

medium
Evaluate ∫ln⁑(2x) dx\int \ln(2x)\,dx.

Example 23

easy
Evaluate ∫xe2x dx\int xe^{2x}\,dx.

Example 24

easy
Evaluate ∫xsin⁑(2x) dx\int x\sin(2x)\,dx.

Example 25

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

Example 26

medium
Evaluate ∫x2cos⁑x dx\int x^2\cos x\,dx.

Example 27

medium
Evaluate ∫(x+1)eβˆ’x dx\int (x+1)e^{-x}\,dx.

Example 28

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

Example 29

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

Example 30

medium
Evaluate ∫xβ‹…2x dx\int x \cdot 2^x\,dx.

Example 31

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

Example 32

medium
Evaluate ∫xx+1 dx\int x\sqrt{x+1}\,dx using parts.

Example 33

medium
Evaluate ∫arcsin⁑x dx\int \arcsin x\,dx.

Example 34

hard
Evaluate ∫e2xsin⁑(3x) dx\int e^{2x}\sin(3x)\,dx.

Example 35

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

Example 36

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

Example 37

hard
Evaluate ∫01x2eβˆ’x dx\int_0^1 x^2 e^{-x}\,dx.

Example 38

hard
Evaluate ∫xarctan⁑x dx\int x\arctan x\,dx.

Example 39

hard
Evaluate ∫sin⁑(ln⁑x) dx\int \sin(\ln x)\,dx.

Example 40

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

Example 41

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

IntegralDerivative

Background Knowledge

These ideas may be useful before you work through the harder examples.

integralderivative