Calculus Examples

\frac{1}{x^{3}}

$\frac{1}{x^{3}}$

Step 1

Apply basic rules of exponents.

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Step 1.1

Move

x^{3}

$x^{3}$ out of the denominator by raising it to the

- 1

$- 1$ power.

\int {(x^{3})}^{- 1} d x

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

Step 1.2

Multiply the exponents in

{(x^{3})}^{- 1}

${(x^{3})}^{- 1}$ .

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Step 1.2.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

\int x^{3 \cdot - 1} d x

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

Step 1.2.2

Multiply

3

$3$ by

- 1

$- 1$ .

\int x^{- 3} d x

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

\int x^{- 3} d x

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

\int x^{- 3} d x

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

Step 2

By the Power Rule, the integral of

x^{- 3}

$x^{- 3}$ with respect to

x

$x$ is

- \frac{1}{2} x^{- 2}

$- \frac{1}{2} x^{- 2}$ .

- \frac{1}{2} x^{- 2} + C

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

Step 3

Simplify the answer.

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Step 3.1

Rewrite

- \frac{1}{2} x^{- 2} + C

$- \frac{1}{2} x^{- 2} + C$ as

- \frac{1}{2} \cdot \frac{1}{x^{2}} + C

$- \frac{1}{2} \cdot \frac{1}{x^{2}} + C$ .

- \frac{1}{2} \cdot \frac{1}{x^{2}} + C

$- \frac{1}{2} \cdot \frac{1}{x^{2}} + C$

Step 3.2

Simplify.

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Step 3.2.1

Multiply

\frac{1}{x^{2}}

$\frac{1}{x^{2}}$ by

\frac{1}{2}

$\frac{1}{2}$ .

- \frac{1}{x^{2} \cdot 2} + C

$- \frac{1}{x^{2} \cdot 2} + C$

Step 3.2.2

Move

2

$2$ to the left of

x^{2}

$x^{2}$ .

- \frac{1}{2 x^{2}} + C

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

- \frac{1}{2 x^{2}} + C

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

- \frac{1}{2 x^{2}} + C

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

[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$