Calculus Examples

$\int - e^{- 2 \csc (2 x)} \cot (2 x) \csc (2 x) d x$

Step 1

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$- \int e^{- 2 \csc (2 x)} \cot (2 x) \csc (2 x) d x$

Step 2

Let $u_{2} = - 2 \csc (2 x)$ . Then $d u_{2} = 4 \cot (2 x) \csc (2 x) d x$ , so $\frac{1}{4} d u_{2} = \cot (2 x) \csc (2 x) d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 2.1

Let $u_{2} = - 2 \csc (2 x)$ . Find $\frac{d u_{2}}{d x}$ .

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Step 2.1.1

Differentiate $- 2 \csc (2 x)$ .

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

Step 2.1.2

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

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

Step 2.1.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \csc (x)$ and $g (x) = 2 x$ .

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Step 2.1.3.1

To apply the Chain Rule, set $u_{1}$ as $2 x$ .

$- 2 (\frac{d}{d u_{1}} [\csc (u_{1})] \frac{d}{d x} [2 x])$

Step 2.1.3.2

The derivative of $\csc (u_{1})$ with respect to $u_{1}$ is $- \csc (u_{1}) \cot (u_{1})$ .

$- 2 (- \csc (u_{1}) \cot (u_{1}) \frac{d}{d x} [2 x])$

Step 2.1.3.3

Replace all occurrences of $u_{1}$ with $2 x$ .

$- 2 (- \csc (2 x) \cot (2 x) \frac{d}{d x} [2 x])$

Step 2.1.4

Differentiate.

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Step 2.1.4.1

Multiply $- 1$ by $- 2$ .

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

Step 2.1.4.2

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

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

Step 2.1.4.3

Multiply $2$ by $2$ .

$4 \csc (2 x) \cot (2 x) \frac{d}{d x} [x]$

Step 2.1.4.4

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

$4 \csc (2 x) \cot (2 x) \cdot 1$

Step 2.1.4.5

Simplify the expression.

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Step 2.1.4.5.1

Multiply $4$ by $1$ .

$4 \csc (2 x) \cot (2 x)$

Step 2.1.4.5.2

Reorder the factors of $4 \csc (2 x) \cot (2 x)$ .

$4 \cot (2 x) \csc (2 x)$

Step 2.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$- \int e^{u_{2}} \frac{1}{4} d u_{2}$

Step 3

Combine $e^{u_{2}}$ and $\frac{1}{4}$ .

$- \int \frac{e^{u_{2}}}{4} d u_{2}$

Step 4

Since $\frac{1}{4}$ is constant with respect to $u_{2}$ , move $\frac{1}{4}$ out of the integral.

$- (\frac{1}{4} \int e^{u_{2}} d u_{2})$

Step 5

The integral of $e^{u_{2}}$ with respect to $u_{2}$ is $e^{u_{2}}$ .

$- \frac{1}{4} (e^{u_{2}} + C)$

Step 6

Simplify.

$- \frac{1}{4} e^{u_{2}} + C$

Step 7

Replace all occurrences of $u_{2}$ with $- 2 \csc (2 x)$ .

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