Calculus Examples

$\int \frac{\tan (\ln (x^{2}))}{4 x} d x$

Step 1

Rewrite $\frac{\tan (\ln (x^{2}))}{4 x}$ as $\frac{\tan (2 \ln (x))}{4 x}$ .

$\int \frac{\tan (2 \ln (x))}{4 x} d x$

Step 2

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int \frac{\tan (2 \ln (x))}{x} d x$

Step 3

Let $u = 2 \ln (x)$ . Then $d u = \frac{2}{x} d x$ , so $- d u = \frac{- 2}{x} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1

Let $u = 2 \ln (x)$ . Find $\frac{d u}{d x}$ .

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Step 3.1.1

Differentiate $2 \ln (x)$ .

$\frac{d}{d x} [2 \ln (x)]$

Step 3.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 \ln (x)$ with respect to $x$ is $2 \frac{d}{d x} [\ln (x)]$ .

$2 \frac{d}{d x} [\ln (x)]$

Step 3.1.3

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

$2 \frac{1}{x}$

Step 3.1.4

Combine $2$ and $\frac{1}{x}$ .

$\frac{2}{x}$

Step 3.2

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

$\frac{1}{4} \int \frac{1}{2} \tan (u) d u$

Step 4

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

$\frac{1}{4} (\frac{1}{2} \int \tan (u) d u)$

Step 5

Simplify.

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Step 5.1

Multiply $\frac{1}{2}$ by $\frac{1}{4}$ .

$\frac{1}{2 \cdot 4} \int \tan (u) d u$

Step 5.2

Multiply $2$ by $4$ .

$\frac{1}{8} \int \tan (u) d u$

Step 6

The integral of $\tan (u)$ with respect to $u$ is $\ln (| \sec (u) |)$ .

$\frac{1}{8} (\ln (| \sec (u) |) + C)$

Step 7

Simplify.

$\frac{1}{8} \ln (| \sec (u) |) + C$

Step 8

Replace all occurrences of $u$ with $2 \ln (x)$ .

$\frac{1}{8} \ln (| \sec (2 \ln (x)) |) + C$