How to Use U-Substitution

The method

Choose $u$ , usually the inside function.
Find $d u$ and convert everything, including $d x$ .
Integrate in terms of $u$ .
Substitute back and add $C$ .

Worked examples

Example: $\int 2 x (x^{2} + 1)^{3} d x$

Let $u = x^{2} + 1$ , so $d u = 2 x d x$ .

$\int u^{3} d u = \frac{u ^{4}}{4} + C$

$\frac{( x ^{2} + 1 ) ^{4}}{4} + C$

Example: $\int cos (3 x) d x$

Let $u = 3 x$ , so $d x = \frac{1}{3} d u$ .

$\frac{1}{3} \int cos u d u = \frac{1}{3} sin (3 x) + C$

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

Let $u = x^{2}$ , so $x d x = \frac{1}{2} d u$ .

$\frac{1}{2} \int e^{u} d u = \frac{1}{2} e^{x^{2}} + C$

Common mistake

$\int (2 x + 1)^{4} d x = \frac{( 2 x + 1 ) ^{5}}{5} + C$

This ignores $d x$ . With $u = 2 x + 1$ , $d x = \frac{1}{2} d u$ , so the answer is $\frac{( 2 x + 1 ) ^{5}}{10} + C$ . Differentiate your answer to check it.

Try it yourself

$\int 3 x^{2} (x^{3} + 1)^{2} d x$
Show answer
$\frac{( x ^{3} + 1 ) ^{3}}{3} + C$
$\int sin (2 x) d x$
Show answer
$- \frac{1}{2} cos (2 x) + C$
$\int \frac{x}{x ^{2} + 1} d x$
Show answer
$\frac{1}{2} ln (x^{2} + 1) + C$

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