Calculus Examples

$\int \frac{- 4 \cos (x)}{1 - \cos^{2} (x)} d x$

Step 1

Simplify.

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Step 1.1

Apply pythagorean identity.

$\int \frac{- 4 \cos (x)}{\sin^{2} (x)} d x$

Step 1.2

Factor $\sin (x)$ out of $\sin^{2} (x)$ .

$\int \frac{- 4 \cos (x)}{\sin (x) \sin (x)} d x$

Step 1.3

Separate fractions.

$\int \frac{- 4}{\sin (x)} \cdot \frac{\cos (x)}{\sin (x)} d x$

Step 1.4

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

$\int \frac{- 4}{\sin (x)} \cot (x) d x$

Step 1.5

Separate fractions.

$\int \frac{- 4}{1} \cdot \frac{1}{\sin (x)} \cot (x) d x$

Step 1.6

Convert from $\frac{1}{\sin (x)}$ to $\csc (x)$ .

$\int \frac{- 4}{1} \csc (x) \cot (x) d x$

Step 1.7

Divide $- 4$ by $1$ .

$\int - 4 \csc (x) \cot (x) d x$

Step 2

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

$- 4 \int \csc (x) \cot (x) d x$

Step 3

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

$- 4 (- \csc (x) + C)$

Step 4

Simplify the answer.

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Step 4.1

Simplify.

$- 4 (- \csc (x)) + C$

Step 4.2

Multiply $- 1$ by $- 4$ .

$4 \csc (x) + C$