Calculus Examples

$x^{- 3}$

Step 1

Write $x^{- 3}$ as a function.

$f (x) = x^{- 3}$

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.

$F (x) = \int x^{- 3} d x$

Step 4

By the Power Rule, the integral of $x^{- 3}$ with respect to $x$ is $- \frac{1}{2} x^{- 2}$ .

$- \frac{1}{2} x^{- 2} + C$

Step 5

Simplify the answer.

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Step 5.1

Rewrite $- \frac{1}{2} x^{- 2} + C$ as $- \frac{1}{2} \cdot \frac{1}{x^{2}} + C$ .

$- \frac{1}{2} \cdot \frac{1}{x^{2}} + C$

Step 5.2

Simplify.

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Step 5.2.1

Multiply $\frac{1}{x^{2}}$ by $\frac{1}{2}$ .

$- \frac{1}{x^{2} \cdot 2} + C$

Step 5.2.2

Move $2$ to the left of $x^{2}$ .

$- \frac{1}{2 x^{2}} + C$

Step 6

The answer is the antiderivative of the function $f (x) = x^{- 3}$ .

$F (x) =$ $- \frac{1}{2 x^{2}} + C$