Calculus Examples

$f (x) = \frac{1}{2 x^{3}}$

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.

$F (x) = \int \frac{1}{2 x^{3}} d x$

Step 3

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int \frac{1}{x^{3}} d x$

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.

$\frac{1}{2} \int {(x^{3})}^{- 1} d x$

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, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{1}{2} \int x^{3 \cdot - 1} d x$

Step 4.2.2

Multiply $3$ by $- 1$ .

$\frac{1}{2} \int x^{- 3} d x$

Step 5

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

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

Step 6

Simplify the answer.

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Step 6.1

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

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

Step 6.2

Simplify.

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Step 6.2.1

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

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

Step 6.2.2

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

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

Step 6.2.3

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

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

Step 6.2.4

Multiply $2$ by $2$ .

$- \frac{1}{4 x^{2}} + C$

Step 7

The answer is the antiderivative of the function $f (x) = \frac{1}{2 x^{3}}$ .

$F (x) =$ $- \frac{1}{4 x^{2}} + C$