Calculus Examples

$\frac{\cos (x)}{\sin^{2} (x)}$

Step 1

Write $\frac{\cos (x)}{\sin^{2} (x)}$ as a function.

$f (x) = \frac{\cos (x)}{\sin^{2} (x)}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \frac{\cos (x)}{\sin^{2} (x)} d x$

Step 4

Let $u = \sin (x)$ . Then $d u = \cos (x) d x$ , so $\frac{1}{\cos (x)} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

Let $u = \sin (x)$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $\sin (x)$ .

$\frac{d}{d x} [\sin (x)]$

Step 4.1.2

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$\cos (x)$

Step 4.2

Rewrite the problem using $u$ and $d u$ .

$\int \frac{1}{u^{2}} d u$

Step 5

Apply basic rules of exponents.

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Step 5.1

Move $u^{2}$ out of the denominator by raising it to the $- 1$ power.

$\int {(u^{2})}^{- 1} d u$

Step 5.2

Multiply the exponents in ${(u^{2})}^{- 1}$ .

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Step 5.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int u^{2 \cdot - 1} d u$

Step 5.2.2

Multiply $2$ by $- 1$ .

$\int u^{- 2} d u$

Step 6

By the Power Rule, the integral of $u^{- 2}$ with respect to $u$ is $- u^{- 1}$ .

$- u^{- 1} + C$

Step 7

Rewrite $- u^{- 1} + C$ as $- \frac{1}{u} + C$ .

$- \frac{1}{u} + C$

Step 8

Replace all occurrences of $u$ with $\sin (x)$ .

$- \frac{1}{\sin (x)} + C$

Step 9

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

$- \csc (x) + C$

Step 10

The answer is the antiderivative of the function $f (x) = \frac{\cos (x)}{\sin^{2} (x)}$ .

$F (x) =$ $- \csc (x) + C$