Calculus Examples

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

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.

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

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

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

Step 1.2.2

Multiply $9$ by $- 1$ .

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

Step 2

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

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

Step 3

Simplify the answer.

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Step 3.1

Simplify.

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Step 3.1.1

Combine $x^{- 8}$ and $\frac{1}{8}$ .

$(- \frac{x^{- 8}}{8} + C) d x$

Step 3.1.2

Move $x^{- 8}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 3.2

Rewrite $(- \frac{1}{8 x^{8}} + C) d x$ as $- \frac{d}{8 x^{8}} x + C$ .

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

Step 3.3

Simplify.

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Step 3.3.1

Combine $x$ and $\frac{d}{8 x^{8}}$ .

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

Step 3.3.2

Cancel the common factor of $x$ and $x^{8}$ .

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Step 3.3.2.1

Factor $x$ out of $x d$ .

$- \frac{x (d)}{8 x^{8}} + C$

Step 3.3.2.2

Cancel the common factors.

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Step 3.3.2.2.1

Factor $x$ out of $8 x^{8}$ .

$- \frac{x (d)}{x (8 x^{7})} + C$

Step 3.3.2.2.2

Cancel the common factor.

$- \frac{x d}{x (8 x^{7})} + C$

Step 3.3.2.2.3

Rewrite the expression.

$- \frac{d}{8 x^{7}} + C$