Calculus Examples

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

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

$- 3 (- \cot (x) + C) + \int 4 \csc (2 x) \cot (2 x) d x$

Step 5

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

$- 3 (- \cot (x) + C) + 4 \int \csc (2 x) \cot (2 x) d x$

Step 6

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

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Step 6.1

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

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Step 6.1.1

Differentiate $\csc (2 x)$ .

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

Step 6.1.2

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 6.1.2.1

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

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

Step 6.1.2.2

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

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

Step 6.1.2.3

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

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

Step 6.1.3

Differentiate.

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Step 6.1.3.1

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

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

Step 6.1.3.2

Multiply $2$ by $- 1$ .

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

Step 6.1.3.3

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

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

Step 6.1.3.4

Simplify the expression.

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Step 6.1.3.4.1

Multiply $- 2$ by $1$ .

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

Step 6.1.3.4.2

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

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

Step 6.2

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

$- 3 (- \cot (x) + C) + 4 \int \frac{1}{- 2} d u_{2}$

Step 7

Move the negative in front of the fraction.

$- 3 (- \cot (x) + C) + 4 \int - \frac{1}{2} d u_{2}$

Step 8

Apply the constant rule.

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

Step 9

Simplify.

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Step 9.1

Simplify.

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

Step 9.2

Simplify.

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Step 9.2.1

Combine $u_{2}$ and $\frac{1}{2}$ .

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

Step 9.2.2

Multiply $- 1$ by $4$ .

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

Step 9.2.3

Combine $- 4$ and $\frac{u_{2}}{2}$ .

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

Step 9.2.4

Cancel the common factor of $- 4$ and $2$ .

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Step 9.2.4.1

Factor $2$ out of $- 4 u_{2}$ .

$3 \cot (x) + \frac{2 (- 2 u_{2})}{2} + C$

Step 9.2.4.2

Cancel the common factors.

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Step 9.2.4.2.1

Factor $2$ out of $2$ .

$3 \cot (x) + \frac{2 (- 2 u_{2})}{2 (1)} + C$

Step 9.2.4.2.2

Cancel the common factor.

$3 \cot (x) + \frac{2 (- 2 u_{2})}{2 \cdot 1} + C$

Step 9.2.4.2.3

Rewrite the expression.

$3 \cot (x) + \frac{- 2 u_{2}}{1} + C$

Step 9.2.4.2.4

Divide $- 2 u_{2}$ by $1$ .

$3 \cot (x) - 2 u_{2} + C$

Step 10

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

$3 \cot (x) - 2 \csc (2 x) + C$