Practice u-Substitution 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 where you substitute u=g(x)u = g(x) and du=g′(x) dxdu = 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 uu, and the integral becomes simpler.

Showing a random 20 of 50 problems.

Example 1

medium
Evaluate ∫(ln⁡x)3x dx\displaystyle\int \frac{(\ln x)^3}{x}\,dx.

Example 2

hard
Evaluate ∫x1+x2 dx\displaystyle\int \frac{x}{\sqrt{1 + x^2}}\,dx.

Example 3

medium
Evaluate ∫012xx2+1 dx\int_0^1 \frac{2x}{x^2+1}\,dx.

Example 4

easy
For ∫2x ex2 dx\int 2x\,e^{x^2}\,dx, what is the natural choice of uu?

Example 5

medium
Evaluate ∫tan⁡x dx\int \tan x\,dx (write tan⁡x=sin⁡xcos⁡x\tan x=\frac{\sin x}{\cos x}).

Example 6

easy
Evaluate ∫sin⁡x cos⁡x dx\int \sin x\,\cos x\,dx using u=sin⁡xu=\sin x.

Example 7

hard
Find ∫ln⁡xx dx\displaystyle\int \frac{\ln x}{x}\,dx.

Example 8

hard
Evaluate ∫1x ln⁡x dx\displaystyle\int \frac{1}{x\,\ln x}\,dx.

Example 9

easy
Evaluate ∫(3x+2)5 dx\int (3x+2)^5\,dx.

Example 10

hard
Evaluate ∫x3x2+1 dx\displaystyle\int x^3 \sqrt{x^2 + 1}\,dx via u=x2+1u = x^2 + 1.

Example 11

easy
Evaluate ∫e−2x dx\displaystyle\int e^{-2x}\,dx.

Example 12

challenge
Evaluate ∫xx+1 dx\int x\sqrt{x+1}\,dx (note: the inner derivative is not present).

Example 13

medium
Evaluate ∫xx2+1 dx\int \frac{x}{x^2+1}\,dx.

Example 14

medium
Evaluate ∫exex+2 dx\displaystyle\int \frac{e^x}{e^x + 2}\,dx.

Example 15

medium
Evaluate ∫sec⁡2x tan⁡3x dx\int \sec^2 x\,\tan^3 x\,dx.

Example 16

medium
Evaluate ∫x2(x3+2)4 dx\int x^2 (x^3+2)^4\,dx.

Example 17

challenge
Evaluate ∫11+x dx\displaystyle\int \frac{1}{1 + \sqrt{x}}\,dx.

Example 18

easy
Evaluate ∫(5x+3)4 dx\displaystyle\int (5x+3)^4\,dx.

Example 19

easy
Evaluate ∫2x(x2+1)3 dx\int 2x(x^2+1)^3\,dx using u=x2+1u=x^2+1.

Example 20

easy
For the definite integral ∫032x(x2) dx\int_0^3 2x(x^2)\,dx with u=x2u=x^2, what are the new limits?
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