Calculus Examples

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

Step 1

Let $u = \frac{x}{2}$ . Then $d u = \frac{1}{2} d x$ , so $2 d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

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

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Step 1.1.1

Differentiate $\frac{x}{2}$ .

$\frac{d}{d x} [\frac{x}{2}]$

Step 1.1.2

Since $\frac{1}{2}$ is constant with respect to $x$ , the derivative of $\frac{x}{2}$ with respect to $x$ is $\frac{1}{2} \frac{d}{d x} [x]$ .

$\frac{1}{2} \frac{d}{d x} [x]$

Step 1.1.3

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

$\frac{1}{2} \cdot 1$

Step 1.1.4

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

$\frac{1}{2}$

Step 1.2

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

$\int \csc^{2} (u) \frac{1}{\frac{1}{2}} d u$

Step 2

Simplify.

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Step 2.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{2}$ .

$\int \csc^{2} (u) (1 \cdot 2) d u$

Step 2.2

Multiply $2$ by $1$ .

$\int \csc^{2} (u) \cdot 2 d u$

Step 2.3

Move $2$ to the left of $\csc^{2} (u)$ .

$\int 2 \csc^{2} (u) d u$

Step 3

Since $2$ is constant with respect to $u$ , move $2$ out of the integral.

$2 \int \csc^{2} (u) d u$

Step 4

Since the derivative of $- \cot (u)$ is $\csc^{2} (u)$ , the integral of $\csc^{2} (u)$ is $- \cot (u)$ .

$2 (- \cot (u) + C)$

Step 5

Simplify.

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Step 5.1

Simplify.

$2 (- \cot (u)) + C$

Step 5.2

Multiply $- 1$ by $2$ .

$- 2 \cot (u) + C$

Step 6

Replace all occurrences of $u$ with $\frac{x}{2}$ .

$- 2 \cot (\frac{x}{2}) + C$

Step 7

Reorder terms.

$- 2 \cot (\frac{1}{2} x) + C$