Calculus Examples

$\int \frac{1}{x^{9}} d x$

Step 1

Apply basic rules of exponents.

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Step 1.1

Move $x^{9}$ out of the denominator by raising it to the $- 1$ power.

$\int {(x^{9})}^{- 1} d x$

Step 1.2

Multiply the exponents in ${(x^{9})}^{- 1}$ .

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Step 1.2.1

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

$\int x^{9 \cdot - 1} d x$

Step 1.2.2

Multiply $9$ by $- 1$ .

$\int x^{- 9} d x$

Step 2

By the Power Rule, the integral of $x^{- 9}$ with respect to $x$ is $- \frac{1}{8} x^{- 8}$ .

$- \frac{1}{8} x^{- 8} + C$

Step 3

Simplify the answer.

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Step 3.1

Rewrite $- \frac{1}{8} x^{- 8} + C$ as $- \frac{1}{8} \cdot \frac{1}{x^{8}} + C$ .

$- \frac{1}{8} \cdot \frac{1}{x^{8}} + C$

Step 3.2

Simplify.

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Step 3.2.1

Multiply $\frac{1}{x^{8}}$ by $\frac{1}{8}$ .

$- \frac{1}{x^{8} \cdot 8} + C$

Step 3.2.2

Move $8$ to the left of $x^{8}$ .

$- \frac{1}{8 x^{8}} + C$