Calculus Examples

$\int \frac{1}{7 x^{4}} d x$

Step 1

Since $\frac{1}{7}$ is constant with respect to $x$ , move $\frac{1}{7}$ out of the integral.

$\frac{1}{7} \int \frac{1}{x^{4}} d x$

Step 2

Apply basic rules of exponents.

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Step 2.1

Move $x^{4}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{7} \int {(x^{4})}^{- 1} d x$

Step 2.2

Multiply the exponents in ${(x^{4})}^{- 1}$ .

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Step 2.2.1

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

$\frac{1}{7} \int x^{4 \cdot - 1} d x$

Step 2.2.2

Multiply $4$ by $- 1$ .

$\frac{1}{7} \int x^{- 4} d x$

Step 3

By the Power Rule, the integral of $x^{- 4}$ with respect to $x$ is $- \frac{1}{3} x^{- 3}$ .

$\frac{1}{7} (- \frac{1}{3} x^{- 3} + C)$

Step 4

Simplify the answer.

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Step 4.1

Rewrite $\frac{1}{7} (- \frac{1}{3} x^{- 3} + C)$ as $\frac{1}{7} (- \frac{1}{3} \cdot \frac{1}{x^{3}}) + C$ .

$\frac{1}{7} (- \frac{1}{3} \cdot \frac{1}{x^{3}}) + C$

Step 4.2

Simplify.

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Step 4.2.1

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

$\frac{1}{7} (- \frac{1}{x^{3} \cdot 3}) + C$

Step 4.2.2

Move $3$ to the left of $x^{3}$ .

$\frac{1}{7} (- \frac{1}{3 \cdot x^{3}}) + C$

Step 4.2.3

Multiply $\frac{1}{7}$ by $\frac{1}{3 x^{3}}$ .

$- \frac{1}{7 (3 x^{3})} + C$

Step 4.2.4

Multiply $3$ by $7$ .

$- \frac{1}{21 x^{3}} + C$