Calculus Examples

$x^{- 4}$

Step 1

Write $x^{- 4}$ as a function.

$f (x) = x^{- 4}$

Step 2

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

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

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

Step 4

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

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

Step 5

Simplify the answer.

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Step 5.1

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

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

Step 5.2

Simplify.

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Step 5.2.1

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

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

Step 5.2.2

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

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

Step 6

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

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