Calculus Examples

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

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.

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

Step 3

Split the single integral into multiple integrals.

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

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.

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

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

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

Step 4.2.2

Multiply $3$ by $- 1$ .

$\int x^{- 3} d x + \int - 2 d x$

Step 5

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

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

Step 6

Apply the constant rule.

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

Step 7

Simplify.

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Step 7.1

Simplify.

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

Step 7.2

Simplify.

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Step 7.2.1

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

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

Step 7.2.2

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

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

Step 8

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

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