Calculus Examples

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

Write $\frac{\csc^{2} (x)}{\cot^{3} (x)}$ as a function.

$f (x) = \frac{\csc^{2} (x)}{\cot^{3} (x)}$

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

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

Set up the integral to solve.

$F (x) = \int \frac{\csc^{2} (x)}{\cot^{3} (x)} d x$

Let $u = \cot (x)$ . Then $d u = - \csc^{2} (x) d x$ , so $- \frac{1}{\csc^{2} (x)} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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

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Differentiate $\cot (x)$ .

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

The derivative of $\cot (x)$ with respect to $x$ is $- \csc^{2} (x)$ .

$- \csc^{2} (x)$

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

$\int \frac{- 1}{u^{3}} d u$

Move the negative in front of the fraction.

$\int - \frac{1}{u^{3}} d u$

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$- \int \frac{1}{u^{3}} d u$

Apply basic rules of exponents.

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Move $u^{3}$ out of the denominator by raising it to the $- 1$ power.

$- \int {(u^{3})}^{- 1} d u$

Multiply the exponents in ${(u^{3})}^{- 1}$ .

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

$- \int u^{3 \cdot - 1} d u$

Multiply $3$ by $- 1$ .

$- \int u^{- 3} d u$

By the Power Rule, the integral of $u^{- 3}$ with respect to $u$ is $- \frac{1}{2} u^{- 2}$ .

$- (- \frac{1}{2} u^{- 2} + C)$

Simplify.

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Simplify.

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Combine $u^{- 2}$ and $\frac{1}{2}$ .

$- (- \frac{u^{- 2}}{2} + C)$

Move $u^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- (- \frac{1}{2 u^{2}} + C)$

Simplify.

$- - \frac{1}{2 u^{2}} + C$

Simplify.

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Multiply $- 1$ by $- 1$ .

$1 \frac{1}{2 u^{2}} + C$

Multiply $\frac{1}{2 u^{2}}$ by $1$ .

$\frac{1}{2 u^{2}} + C$

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

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

The answer is the antiderivative of the function $f (x) = \frac{\csc^{2} (x)}{\cot^{3} (x)}$ .

$F (x) =$ $\frac{1}{2 \cot^{2} (x)} + C$