u-Substitution Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of u-Substitution.

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 where you substitute $u = g(x)$ and $du = g'(x)\,dx$ to transform a complicated integral into a simpler one. It is the reverse of the chain rule for differentiation.

When you see a composite function inside an integral along with its inner derivative lurking nearby, substitution collapses the composition into a single variable. It's like un-nesting a function: replace the inner part with $u$ , and the integral becomes simpler.

Read the full concept explanation →

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: u-Substitution sets $u=g(x)$ so $du=g'(x)\,dx$ , collapsing a composite-times-inner-derivative integral into a simple one in $u$ .

Common stuck point: The procedure for u-substitution is the easy part; the trap is forgetting to convert $dx$ via $du=g'(x)\,dx$ . Asking "Is there an inner function $g(x)$ inside, with its derivative $g'(x)$ also present as a factor?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is there an inner function $g(x)$ inside, with its derivative $g'(x)$ also present as a factor?

Worked Examples

Example 1

easy

Find

\displaystyle\int 3x^2(x^3+1)^4\,dx

Answer

\frac{(x^3+1)^5}{5} + C

First step

Let

u = x^3+1

, so

du = 3x^2\,dx

Full solution

2
Integral becomes $\int u^4\,du = \frac{u^5}{5} + C$ .
3
Substitute back: $\frac{(x^3+1)^5}{5} + C$ .

3x^2\,dx

is exactly

du

, so the substitution is perfect. After substitution the integral reduces to a simple power rule.

Example 2

medium

Evaluate

\displaystyle\int_0^1 xe^{x^2}\,dx

Example 3

medium

Evaluate

\displaystyle\int x^2 \sqrt{x^3+1}\,dx

Example 4

medium

Evaluate

\displaystyle\int x\,\sqrt[3]{x^2 + 5}\,dx

Practice Problems

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

Example 1

easy

Find

\displaystyle\int \cos(5x)\,dx

Example 2

hard

Find

\displaystyle\int \frac{\ln x}{x}\,dx

Example 3

easy

Evaluate

\int 2x(x^2+1)^3\,dx

using

u=x^2+1

Example 4

easy

Evaluate

\int \cos(3x)\,dx

Example 5

easy

Evaluate

\int e^{5x}\,dx

Example 6

easy

Evaluate

\int \frac{1}{2x+1}\,dx

Example 7

easy

Evaluate

\int (3x+2)^5\,dx

Example 8

easy

Evaluate

\int 2x\,e^{x^2}\,dx

Example 9

easy

Evaluate

\int \sin x\,\cos x\,dx

using

u=\sin x

Example 10

easy

For the definite integral

\int_0^3 2x(x^2)\,dx

with

u=x^2

, what are the new limits?

Example 11

medium

Evaluate

\int x\sqrt{x^2+4}\,dx

Example 12

medium

Evaluate the definite integral

\int_0^2 x(x^2+1)^3\,dx

Example 13

medium

Evaluate

\int \frac{\ln x}{x}\,dx

Example 14

medium

Evaluate

\int \frac{x}{x^2+1}\,dx

Example 15

medium

Evaluate

\int \sec^2 x\,\tan^3 x\,dx

Example 16

medium

Evaluate

\int x^2 (x^3+2)^4\,dx

Example 17

challenge

Evaluate

\int_0^{\pi/2} \sin^2 x\,\cos x\,dx

Example 18

challenge

Evaluate

\int x\sqrt{x+1}\,dx

(note: the inner derivative is not present).

Example 19

challenge

Evaluate

\int \frac{1}{\sqrt{x}\,(1+\sqrt{x})}\,dx

Example 20

medium

Evaluate

\int \frac{6x}{(3x^2+1)^2}\,dx

Example 21

medium

Evaluate

\int \tan x\,dx

(write

\tan x=\frac{\sin x}{\cos x}

Example 22

medium

Evaluate

\int_0^1 \frac{2x}{x^2+1}\,dx

Example 23

easy

Evaluate

\displaystyle\int 4x^3(x^4+1)^2\,dx

Example 24

easy

Evaluate

\displaystyle\int \sin(7x)\,dx

Example 25

easy

Evaluate

\displaystyle\int e^{-2x}\,dx

Example 26

easy

Evaluate

\displaystyle\int \frac{1}{3x-5}\,dx

Example 27

easy

Evaluate

\displaystyle\int (5x+3)^4\,dx

Example 28

medium

Evaluate

\displaystyle\int \cos x\,\sin^4 x\,dx

Example 29

medium

Evaluate

\displaystyle\int \frac{2x}{x^2 + 4}\,dx

Example 30

medium

Evaluate

\displaystyle\int_0^1 x(x^2+1)^4\,dx

Example 31

medium

Evaluate

\displaystyle\int \sec^2(3x)\,dx

Example 32

medium

Evaluate

\displaystyle\int \frac{e^x}{e^x + 2}\,dx

Example 33

medium

Evaluate

\displaystyle\int \cos^3 x\,dx

by writing

\cos^3 x = (1 - \sin^2 x)\cos x

and substituting

u = \sin x

Example 34

medium

Evaluate

\displaystyle\int_0^{\pi/4} \tan x \sec^2 x\,dx

Example 35

medium

Evaluate

\displaystyle\int \frac{(\ln x)^3}{x}\,dx

Example 36

medium

Evaluate

\displaystyle\int e^{\sin x}\cos x\,dx

Example 37

hard

Evaluate

\displaystyle\int_0^2 x\sqrt{4 - x^2}\,dx

Example 38

hard

Evaluate

\displaystyle\int \frac{x}{\sqrt{1 + x^2}}\,dx

Example 39

hard

Evaluate

\displaystyle\int x^3 \sqrt{x^2 + 1}\,dx

via

u = x^2 + 1

Example 40

hard

Evaluate

\displaystyle\int \frac{x^2}{(x^3 + 2)^4}\,dx

Example 41

hard

Evaluate

\displaystyle\int \frac{\sin x}{1 + \cos x}\,dx

Example 42

hard

Evaluate

\displaystyle\int_0^{\ln 2} \frac{e^x}{1 + e^x}\,dx

Example 43

hard

Evaluate

\displaystyle\int \frac{1}{x\,\ln x}\,dx

Example 44

challenge

Evaluate

\displaystyle\int \sin^5 x\,dx

by writing

\sin^5 x = (1 - \cos^2 x)^2 \sin x

and substituting

u = \cos x

Example 45

challenge

Evaluate

\displaystyle\int \frac{1}{1 + \sqrt{x}}\,dx

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Related Concepts

Integral Chain Rule Integration by Parts

Background Knowledge

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

integralchain rule