Calculus Examples

$2 \csc^{2} (\frac{x}{3})$

Step 1

Write $2 \csc^{2} (\frac{x}{3})$ as a function.

$f (x) = 2 \csc^{2} (\frac{x}{3})$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int 2 \csc^{2} (\frac{x}{3}) d x$

Step 4

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

$2 \int \csc^{2} (\frac{x}{3}) d x$

Step 5

Let $u = \frac{x}{3}$ . Then $d u = \frac{1}{3} d x$ , so $3 d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = \frac{x}{3}$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1

Differentiate $\frac{x}{3}$ .

$\frac{d}{d x} [\frac{x}{3}]$

Step 5.1.2

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{x}{3}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [x]$ .

$\frac{1}{3} \frac{d}{d x} [x]$

Step 5.1.3

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

$\frac{1}{3} \cdot 1$

Step 5.1.4

Multiply $\frac{1}{3}$ by $1$ .

$\frac{1}{3}$

Step 5.2

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

$2 \int \csc^{2} (u) \frac{1}{\frac{1}{3}} d u$

Step 6

Simplify.

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Step 6.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{3}$ .

$2 \int \csc^{2} (u) (1 \cdot 3) d u$

Step 6.2

Multiply $3$ by $1$ .

$2 \int \csc^{2} (u) \cdot 3 d u$

Step 6.3

Move $3$ to the left of $\csc^{2} (u)$ .

$2 \int 3 \csc^{2} (u) d u$

Step 7

Since $3$ is constant with respect to $u$ , move $3$ out of the integral.

$2 (3 \int \csc^{2} (u) d u)$

Step 8

Multiply $3$ by $2$ .

$6 \int \csc^{2} (u) d u$

Step 9

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

$6 (- \cot (u) + C)$

Step 10

Simplify.

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Step 10.1

Simplify.

$6 (- \cot (u)) + C$

Step 10.2

Multiply $- 1$ by $6$ .

$- 6 \cot (u) + C$

Step 11

Replace all occurrences of $u$ with $\frac{x}{3}$ .

$- 6 \cot (\frac{x}{3}) + C$

Step 12

Reorder terms.

$- 6 \cot (\frac{1}{3} x) + C$

Step 13

The answer is the antiderivative of the function $f (x) = 2 \csc^{2} (\frac{x}{3})$ .

$F (x) =$ $- 6 \cot (\frac{1}{3} x) + C$