Calculus Examples

$\int \frac{1}{x^{6}} d x d x$

Step 1

Apply basic rules of exponents.

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Step 1.1

Move $x^{6}$ out of the denominator by raising it to the $- 1$ power.

$\int {(x^{6})}^{- 1} d x d x$

Step 1.2

Multiply the exponents in ${(x^{6})}^{- 1}$ .

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Step 1.2.1

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

$\int x^{6 \cdot - 1} d x d x$

Step 1.2.2

Multiply $6$ by $- 1$ .

$\int x^{- 6} d x d x$

Step 2

By the Power Rule, the integral of $x^{- 6}$ with respect to $x$ is $- \frac{1}{5} x^{- 5}$ .

$(- \frac{1}{5} x^{- 5} + C) d x$

Step 3

Simplify the answer.

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Step 3.1

Simplify.

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Step 3.1.1

Combine $x^{- 5}$ and $\frac{1}{5}$ .

$(- \frac{x^{- 5}}{5} + C) d x$

Step 3.1.2

Move $x^{- 5}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$(- \frac{1}{5 x^{5}} + C) d x$

Step 3.2

Rewrite $(- \frac{1}{5 x^{5}} + C) d x$ as $- \frac{d}{5 x^{5}} x + C$ .

$- \frac{d}{5 x^{5}} x + C$

Step 3.3

Simplify.

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Step 3.3.1

Combine $x$ and $\frac{d}{5 x^{5}}$ .

$- \frac{x d}{5 x^{5}} + C$

Step 3.3.2

Cancel the common factor of $x$ and $x^{5}$ .

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Step 3.3.2.1

Factor $x$ out of $x d$ .

$- \frac{x (d)}{5 x^{5}} + C$

Step 3.3.2.2

Cancel the common factors.

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Step 3.3.2.2.1

Factor $x$ out of $5 x^{5}$ .

$- \frac{x (d)}{x (5 x^{4})} + C$

Step 3.3.2.2.2

Cancel the common factor.

$- \frac{x d}{x (5 x^{4})} + C$

Step 3.3.2.2.3

Rewrite the expression.

$- \frac{d}{5 x^{4}} + C$