Calculus Examples

$\int \csc^{4} (\frac{x}{4}) d x$

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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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.1.4

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

$\frac{1}{4}$

Step 1.2

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

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

Step 2

Simplify.

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Step 2.1

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

$\int \csc^{4} (u) (1 \cdot 4) d u$

Step 2.2

Multiply $4$ by $1$ .

$\int \csc^{4} (u) \cdot 4 d u$

Step 2.3

Move $4$ to the left of $\csc^{4} (u)$ .

$\int 4 \csc^{4} (u) d u$

Step 3

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

$4 \int \csc^{4} (u) d u$

Step 4

Apply the reduction formula.

$4 (- \frac{\cot (u) \csc^{2} (u)}{3} + \frac{2}{3} \int \csc^{2} (u) d u)$

Step 5

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

$4 (- \frac{\cot (u) \csc^{2} (u)}{3} + \frac{2}{3} (- \cot (u) + C))$

Step 6

Rewrite $4 (- \frac{\cot (u) \csc^{2} (u)}{3} + \frac{2}{3} (- \cot (u) + C))$ as $4 (- \frac{\cot (u) \csc^{2} (u)}{3} - \frac{2 \cot (u)}{3}) + C$ .

$4 (- \frac{\cot (u) \csc^{2} (u)}{3} - \frac{2 \cot (u)}{3}) + C$

Step 7

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

$4 (- \frac{\cot (\frac{x}{4}) \csc^{2} (\frac{x}{4})}{3} - \frac{2 \cot (\frac{x}{4})}{3}) + C$

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

$4 (- \frac{\cot (\frac{x}{4}) \csc^{2} (\frac{x}{4})}{3}) + 4 (- \frac{2 \cot (\frac{x}{4})}{3}) + C$

Step 8.2

Multiply $4 (- \frac{\cot (\frac{x}{4}) \csc^{2} (\frac{x}{4})}{3})$ .

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Step 8.2.1

Multiply $- 1$ by $4$ .

$- 4 \frac{\cot (\frac{x}{4}) \csc^{2} (\frac{x}{4})}{3} + 4 (- \frac{2 \cot (\frac{x}{4})}{3}) + C$

Step 8.2.2

Combine $- 4$ and $\frac{\cot (\frac{x}{4}) \csc^{2} (\frac{x}{4})}{3}$ .

$\frac{- 4 (\cot (\frac{x}{4}) \csc^{2} (\frac{x}{4}))}{3} + 4 (- \frac{2 \cot (\frac{x}{4})}{3}) + C$

Step 8.3

Multiply $4 (- \frac{2 \cot (\frac{x}{4})}{3})$ .

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Step 8.3.1

Multiply $- 1$ by $4$ .

$\frac{- 4 (\cot (\frac{x}{4}) \csc^{2} (\frac{x}{4}))}{3} - 4 \frac{2 \cot (\frac{x}{4})}{3} + C$

Step 8.3.2

Combine $- 4$ and $\frac{2 \cot (\frac{x}{4})}{3}$ .

$\frac{- 4 (\cot (\frac{x}{4}) \csc^{2} (\frac{x}{4}))}{3} + \frac{- 4 (2 \cot (\frac{x}{4}))}{3} + C$

Step 8.3.3

Multiply $2$ by $- 4$ .

$\frac{- 4 (\cot (\frac{x}{4}) \csc^{2} (\frac{x}{4}))}{3} + \frac{- 8 \cot (\frac{x}{4})}{3} + C$

Step 8.4

Simplify each term.

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Step 8.4.1

Move the negative in front of the fraction.

$- \frac{4 \cot (\frac{x}{4}) \csc^{2} (\frac{x}{4})}{3} + \frac{- 8 \cot (\frac{x}{4})}{3} + C$

Step 8.4.2

Move the negative in front of the fraction.

$- \frac{4 \cot (\frac{x}{4}) \csc^{2} (\frac{x}{4})}{3} - \frac{8 \cot (\frac{x}{4})}{3} + C$

Step 9

Reorder terms.

$- \frac{4}{3} \cot (\frac{1}{4} x) \csc^{2} (\frac{1}{4} x) - \frac{8}{3} \cot (\frac{1}{4} x) + C$