Calculus Examples

$\int \frac{1}{3 x^{6}} d x$

Step 1

Since $\frac{1}{3}$ is constant with respect to $x$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{3} \int \frac{1}{x^{6}} d x$

Step 2

Apply basic rules of exponents.

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Step 2.1

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

$\frac{1}{3} \int {(x^{6})}^{- 1} d x$

Step 2.2

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

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Step 2.2.1

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

$\frac{1}{3} \int x^{6 \cdot - 1} d x$

Step 2.2.2

Multiply $6$ by $- 1$ .

$\frac{1}{3} \int x^{- 6} d x$

Step 3

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

$\frac{1}{3} (- \frac{1}{5} x^{- 5} + C)$

Step 4

Simplify the answer.

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Step 4.1

Rewrite $\frac{1}{3} (- \frac{1}{5} x^{- 5} + C)$ as $\frac{1}{3} (- \frac{1}{5} \cdot \frac{1}{x^{5}}) + C$ .

$\frac{1}{3} (- \frac{1}{5} \cdot \frac{1}{x^{5}}) + C$

Step 4.2

Simplify.

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Step 4.2.1

Multiply $\frac{1}{x^{5}}$ by $\frac{1}{5}$ .

$\frac{1}{3} (- \frac{1}{x^{5} \cdot 5}) + C$

Step 4.2.2

Move $5$ to the left of $x^{5}$ .

$\frac{1}{3} (- \frac{1}{5 \cdot x^{5}}) + C$

Step 4.2.3

Multiply $\frac{1}{3}$ by $\frac{1}{5 x^{5}}$ .

$- \frac{1}{3 (5 x^{5})} + C$

Step 4.2.4

Multiply $5$ by $3$ .

$- \frac{1}{15 x^{5}} + C$