Calculus Examples

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

Step 1

Simplify with factoring out.

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Step 1.1

Factor $2$ out of $4$ .

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

Step 1.2

Rewrite ${\cot (x)}^{2 (2)}$ as exponentiation.

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

Step 2

Using the Pythagorean Identity, rewrite $\cot^{2} (x)$ as $- 1 + \csc^{2} (x)$ .

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

Step 3

Simplify.

$\int 1 - 2 \csc^{2} (x) + \csc^{4} (x) d x$

Step 4

Split the single integral into multiple integrals.

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

Step 5

Apply the constant rule.

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

Step 6

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

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

Step 7

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

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

Step 8

Apply the reduction formula.

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

Step 9

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

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

Step 10

Simplify.

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Step 10.1

Simplify.

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

Step 10.2

Simplify.

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Step 10.2.1

Combine $\frac{2}{3}$ and $\cot (x)$ .

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

Step 10.2.2

To write $2 \cot (x)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 10.2.3

Combine $2 \cot (x)$ and $\frac{3}{3}$ .

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

Step 10.2.4

Combine the numerators over the common denominator.

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

Step 10.2.5

Multiply $3$ by $2$ .

$x - \frac{\cot (x) \csc^{2} (x)}{3} + \frac{6 \cot (x) - 2 \cot (x)}{3} + C$

Step 10.2.6

Subtract $2 \cot (x)$ from $6 \cot (x)$ .

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

Step 11

Reorder terms.

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