Calculus Examples

\int \csc^{3} (x) d x

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

Step 1

Apply the reduction formula.

- \frac{\cot (x) \csc (x)}{2} + \frac{1}{2} \int \csc (x) d x

$- \frac{\cot (x) \csc (x)}{2} + \frac{1}{2} \int \csc (x) d x$

Step 2

The integral of

\csc (x)

$\csc (x)$ with respect to

x

$x$ is

\ln (| \csc (x) - \cot (x) |)

$\ln (| \csc (x) - \cot (x) |)$ .

- \frac{\cot (x) \csc (x)}{2} + \frac{1}{2} (\ln (| \csc (x) - \cot (x) |) + C)

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

Step 3

Simplify the answer.

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Step 3.1

Rewrite

- \frac{\cot (x) \csc (x)}{2} + \frac{1}{2} (\ln (| \csc (x) - \cot (x) |) + C)

$- \frac{\cot (x) \csc (x)}{2} + \frac{1}{2} (\ln (| \csc (x) - \cot (x) |) + C)$ as

- \frac{1}{2} \cot (x) \csc (x) + \frac{1}{2} \ln (| \csc (x) - \cot (x) |) + C

$- \frac{1}{2} \cot (x) \csc (x) + \frac{1}{2} \ln (| \csc (x) - \cot (x) |) + C$ .

- \frac{1}{2} \cot (x) \csc (x) + \frac{1}{2} \ln (| \csc (x) - \cot (x) |) + C

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

Step 3.2

Simplify.

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Step 3.2.1

Combine

\cot (x)

$\cot (x)$ and

\frac{1}{2}

$\frac{1}{2}$ .

- \frac{\cot (x)}{2} \csc (x) + \frac{1}{2} \ln (| \csc (x) - \cot (x) |) + C

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

Step 3.2.2

Combine

\csc (x)

$\csc (x)$ and

\frac{\cot (x)}{2}

$\frac{\cot (x)}{2}$ .

- \frac{\csc (x) \cot (x)}{2} + \frac{1}{2} \ln (| \csc (x) - \cot (x) |) + C

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

Step 3.2.3

To write

\frac{1}{2} \ln (| \csc (x) - \cot (x) |)

$\frac{1}{2} \ln (| \csc (x) - \cot (x) |)$ as a fraction with a common denominator, multiply by

\frac{2}{2}

$\frac{2}{2}$ .

- \frac{\csc (x) \cot (x)}{2} + \frac{1}{2} \ln (| \csc (x) - \cot (x) |) \cdot \frac{2}{2} + C

$- \frac{\csc (x) \cot (x)}{2} + \frac{1}{2} \ln (| \csc (x) - \cot (x) |) \cdot \frac{2}{2} + C$

Step 3.2.4

Combine

\frac{1}{2} \ln (| \csc (x) - \cot (x) |)

$\frac{1}{2} \ln (| \csc (x) - \cot (x) |)$ and

\frac{2}{2}

$\frac{2}{2}$ .

- \frac{\csc (x) \cot (x)}{2} + \frac{\frac{1}{2} \ln (| \csc (x) - \cot (x) |) \cdot 2}{2} + C

$- \frac{\csc (x) \cot (x)}{2} + \frac{\frac{1}{2} \ln (| \csc (x) - \cot (x) |) \cdot 2}{2} + C$

Step 3.2.5

Combine the numerators over the common denominator.

\frac{- \csc (x) \cot (x) + \frac{1}{2} \ln (| \csc (x) - \cot (x) |) \cdot 2}{2} + C

$\frac{- \csc (x) \cot (x) + \frac{1}{2} \ln (| \csc (x) - \cot (x) |) \cdot 2}{2} + C$

Step 3.2.6

Combine

\frac{1}{2}

$\frac{1}{2}$ and

\ln (| \csc (x) - \cot (x) |)

$\ln (| \csc (x) - \cot (x) |)$ .

\frac{- \csc (x) \cot (x) + \frac{\ln (| \csc (x) - \cot (x) |)}{2} \cdot 2}{2} + C

$\frac{- \csc (x) \cot (x) + \frac{\ln (| \csc (x) - \cot (x) |)}{2} \cdot 2}{2} + C$

Step 3.2.7

Combine

\frac{\ln (| \csc (x) - \cot (x) |)}{2}

$\frac{\ln (| \csc (x) - \cot (x) |)}{2}$ and

2

$2$ .

\frac{- \csc (x) \cot (x) + \frac{\ln (| \csc (x) - \cot (x) |) \cdot 2}{2}}{2} + C

$\frac{- \csc (x) \cot (x) + \frac{\ln (| \csc (x) - \cot (x) |) \cdot 2}{2}}{2} + C$

Step 3.2.8

Move

2

$2$ to the left of

\ln (| \csc (x) - \cot (x) |)

$\ln (| \csc (x) - \cot (x) |)$ .

\frac{- \csc (x) \cot (x) + \frac{2 \cdot \ln (| \csc (x) - \cot (x) |)}{2}}{2} + C

$\frac{- \csc (x) \cot (x) + \frac{2 \cdot \ln (| \csc (x) - \cot (x) |)}{2}}{2} + C$

Step 3.2.9

Cancel the common factor of

2

$2$ .

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Step 3.2.9.1

Cancel the common factor.

\frac{- \csc (x) \cot (x) + \frac{2 \ln (| \csc (x) - \cot (x) |)}{2}}{2} + C

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

Step 3.2.9.2

Divide

\ln (| \csc (x) - \cot (x) |)

$\ln (| \csc (x) - \cot (x) |)$ by

1

$1$ .

\frac{- \csc (x) \cot (x) + \ln (| \csc (x) - \cot (x) |)}{2} + C

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

\frac{- \csc (x) \cot (x) + \ln (| \csc (x) - \cot (x) |)}{2} + C

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

\frac{- \csc (x) \cot (x) + \ln (| \csc (x) - \cot (x) |)}{2} + C

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

Step 3.3

Reorder terms.

\frac{1}{2} (- \csc (x) \cot (x) + \ln (| \csc (x) - \cot (x) |)) + C

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

\frac{1}{2} (- \csc (x) \cot (x) + \ln (| \csc (x) - \cot (x) |)) + C

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