Calculus Examples

$\int \cot (\frac{x}{4}) d x$

Step 1

Let $u = \frac{x}{4}$ . Then $d u = \frac{1}{4} d x$ , so $4 d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Let $u = \frac{x}{4}$ . Find $\frac{d u}{d x}$ .

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Differentiate $\frac{x}{4}$ .

$\frac{d}{d x} [\frac{x}{4}]$

Since $\frac{1}{4}$ is constant with respect to $x$ , the derivative of $\frac{x}{4}$ with respect to $x$ is $\frac{1}{4} \frac{d}{d x} [x]$ .

$\frac{1}{4} \frac{d}{d x} [x]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$\frac{1}{4} \cdot 1$

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

$\frac{1}{4}$

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

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

Step 2

Simplify.

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Multiply by the reciprocal of the fraction to divide by $\frac{1}{4}$ .

$\int \cot (u) (1 \cdot 4) d u$

Multiply $4$ by $1$ .

$\int \cot (u) \cdot 4 d u$

Move $4$ to the left of $\cot (u)$ .

$\int 4 \cot (u) d u$

Step 3

Since $4$ is constant with respect to $u$ , move $4$ out of the integral.

$4 \int \cot (u) d u$

Step 4

The integral of $\cot (u)$ with respect to $u$ is $\ln (| \sin (u) |)$ .

$4 (\ln (| \sin (u) |) + C)$

Step 5

Simplify.

$4 \ln (| \sin (u) |) + C$

Step 6

Replace all occurrences of $u$ with $\frac{x}{4}$ .

$4 \ln (| \sin (\frac{x}{4}) |) + C$

Step 7

Reorder terms.

$4 \ln (| \sin (\frac{1}{4} x) |) + C$