Calculus Examples

$\int \frac{1}{x (1 + \ln^{2} (x))} d x$

Step 1

Let $u = \ln (x)$ . Then $d u = \frac{1}{x} d x$ , so $x d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = \ln (x)$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $\ln (x)$ .

$\frac{d}{d x} [\ln (x)]$

Step 1.1.2

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$\frac{1}{x}$

Step 1.2

Rewrite the problem using $u$ and $d u$ .

$\int \frac{1}{1 + u^{2}} d u$

Step 2

Rewrite $1$ as $1^{2}$ .

$\int \frac{1}{1^{2} + u^{2}} d u$

Step 3

The integral of $\frac{1}{1^{2} + u^{2}}$ with respect to $u$ is $\arctan (u) + C$ .

$\arctan (u) + C$

Step 4

Replace all occurrences of $u$ with $\ln (x)$ .

$\arctan (\ln (x)) + C$