Calculus Examples

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

Step 1

Apply the reduction formula.

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

Step 2

The integral of $\csc (x)$ with respect to $x$ is $\ln (| \csc (x) - \cot (x) |)$ .

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

Step 3

Simplify the answer.

Tap for more steps...

Step 3.1

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

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

Step 3.2

Simplify.

Tap for more steps...

Step 3.2.1

Combine $\cot (x)$ and $\frac{1}{2}$ .

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

Step 3.2.2

Combine $\csc (x)$ and $\frac{\cot (x)}{2}$ .

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

Step 3.2.3

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

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

Step 3.2.4

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

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

Step 3.2.5

Combine the numerators over the common denominator.

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

Step 3.2.6

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

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

Step 3.2.7

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

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

Step 3.2.8

Move $2$ to the left of $\ln (| \csc (x) - \cot (x) |)$ .

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

Step 3.2.9

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.9.1

Cancel the common factor.

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

Step 3.2.9.2

Divide $\ln (| \csc (x) - \cot (x) |)$ by $1$ .

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

Step 3.3

Reorder terms.

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