Calculus Examples

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

Step 1

Apply the reduction formula.

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

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

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

Step 3.2

Reorder terms.

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