Calculus Examples

$\int \frac{1}{3 x^{5}} d x$

Step 1

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

$\frac{1}{3} \int \frac{1}{x^{5}} d x$

Step 2

Apply basic rules of exponents.

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Step 2.1

Move $x^{5}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{3} \int {(x^{5})}^{- 1} d x$

Step 2.2

Multiply the exponents in ${(x^{5})}^{- 1}$ .

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Step 2.2.1

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

$\frac{1}{3} \int x^{5 \cdot - 1} d x$

Step 2.2.2

Multiply $5$ by $- 1$ .

$\frac{1}{3} \int x^{- 5} d x$

Step 3

By the Power Rule, the integral of $x^{- 5}$ with respect to $x$ is $- \frac{1}{4} x^{- 4}$ .

$\frac{1}{3} (- \frac{1}{4} x^{- 4} + C)$

Step 4

Simplify the answer.

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Step 4.1

Rewrite $\frac{1}{3} (- \frac{1}{4} x^{- 4} + C)$ as $\frac{1}{3} (- \frac{1}{4} \cdot \frac{1}{x^{4}}) + C$ .

$\frac{1}{3} (- \frac{1}{4} \cdot \frac{1}{x^{4}}) + C$

Step 4.2

Simplify.

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Step 4.2.1

Multiply $\frac{1}{x^{4}}$ by $\frac{1}{4}$ .

$\frac{1}{3} (- \frac{1}{x^{4} \cdot 4}) + C$

Step 4.2.2

Move $4$ to the left of $x^{4}$ .

$\frac{1}{3} (- \frac{1}{4 \cdot x^{4}}) + C$

Step 4.2.3

Multiply $\frac{1}{3}$ by $\frac{1}{4 x^{4}}$ .

$- \frac{1}{3 (4 x^{4})} + C$

Step 4.2.4

Multiply $4$ by $3$ .

$- \frac{1}{12 x^{4}} + C$