Algebra Examples

$\frac{1}{x^{3}}$

Step 1

Write $\frac{1}{x^{3}}$ as a function.

$f (x) = \frac{1}{x^{3}}$

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.

$F (x) = \int \frac{1}{x^{3}} d x$

Step 4

Apply basic rules of exponents.

Tap for more steps...

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

$\int {(x^{3})}^{- 1} d x$

Multiply the exponents in ${(x^{3})}^{- 1}$ .

Tap for more steps...

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

$\int x^{3 \cdot - 1} d x$

Multiply $3$ by $- 1$ .

$\int x^{- 3} d x$

Step 5

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 6

Simplify the answer.

Tap for more steps...

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

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

Simplify.

Tap for more steps...

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

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

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

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

Step 7

The answer is the antiderivative of the function $f (x) = \frac{1}{x^{3}}$ .

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