Calculus Examples

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

Step 1

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 2

Set up the integral to solve.

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

Step 3

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 4

Apply basic rules of exponents.

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

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

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

Step 4.2.2

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.

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Step 6.1

Simplify.

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Step 6.1.1

Combine $x^{- 2}$ and $\frac{1}{2}$ .

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

Step 6.1.2

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 6.2

Simplify.

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

Step 6.3

Simplify.

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Step 6.3.1

Multiply $- 1$ by $- 1$ .

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

Step 6.3.2

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

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

Step 7

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

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