Calculus Examples

$g (x) = - 4 x^{- 5}$

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.

$G (x) = \int - 4 x^{- 5} d x$

Step 3

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$- 4 \int x^{- 5} d x$

Step 4

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

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

Step 5

Simplify the answer.

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Step 5.1

Simplify.

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Step 5.1.1

Combine $x^{- 4}$ and $\frac{1}{4}$ .

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

Step 5.1.2

Move $x^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 5.2

Simplify.

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

Step 5.3

Simplify.

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Step 5.3.1

Multiply $- 1$ by $- 4$ .

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

Step 5.3.2

Combine $4$ and $\frac{1}{4 x^{4}}$ .

$\frac{4}{4 x^{4}} + C$

Step 5.3.3

Cancel the common factor of $4$ .

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Step 5.3.3.1

Cancel the common factor.

$\frac{4}{4 x^{4}} + C$

Step 5.3.3.2

Rewrite the expression.

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

Step 6

The answer is the antiderivative of the function $g (x) = - 4 x^{- 5}$ .

$G (x) =$ $\frac{1}{x^{4}} + C$