Calculus Examples

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

Step 1

Rewrite $\frac{1}{x \ln (x^{2})}$ as $\frac{1}{x (2 \ln (x))}$ .

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

Step 2

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

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

Step 3

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 3.1

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

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Step 3.1.1

Differentiate $\ln (x)$ .

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

Step 3.1.2

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

$\frac{1}{x}$

Step 3.2

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

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

Step 4

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$\frac{1}{2} (\ln (| u |) + C)$

Step 5

Simplify.

$\frac{1}{2} \ln (| u |) + C$

Step 6

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

$\frac{1}{2} \ln (| \ln (x) |) + C$