Calculus Examples

$g (x) = \frac{1}{x^{7}}$

Step 1

The function $G (x)$ can be found by finding the indefinite integral of the derivative $g (x)$ .

$G (x) = \int g (x) d x$

Step 2

Set up the integral to solve.

$G (x) = \int \frac{1}{x^{7}} d x$

Step 3

Apply basic rules of exponents.

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Step 3.1

Move $x^{7}$ out of the denominator by raising it to the $- 1$ power.

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

Step 3.2

Multiply the exponents in ${(x^{7})}^{- 1}$ .

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Step 3.2.1

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

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

Step 3.2.2

Multiply $7$ by $- 1$ .

$\int x^{- 7} d x$

Step 4

By the Power Rule, the integral of $x^{- 7}$ with respect to $x$ is $- \frac{1}{6} x^{- 6}$ .

$- \frac{1}{6} x^{- 6} + C$

Step 5

Simplify the answer.

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Step 5.1

Rewrite $- \frac{1}{6} x^{- 6} + C$ as $- \frac{1}{6} \cdot \frac{1}{x^{6}} + C$ .

$- \frac{1}{6} \cdot \frac{1}{x^{6}} + C$

Step 5.2

Simplify.

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Step 5.2.1

Multiply $\frac{1}{x^{6}}$ by $\frac{1}{6}$ .

$- \frac{1}{x^{6} \cdot 6} + C$

Step 5.2.2

Move $6$ to the left of $x^{6}$ .

$- \frac{1}{6 x^{6}} + C$

Step 6

The answer is the antiderivative of the function $g (x) = \frac{1}{x^{7}}$ .

$G (x) =$ $- \frac{1}{6 x^{6}} + C$