Calculus Examples

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

Step 1

Write $\frac{1}{x^{2}} - x^{2} - \frac{1}{3}$ as a function.

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

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.

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

Apply basic rules of exponents.

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Step 5.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

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

Step 5.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 5.2.1

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

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

Step 5.2.2

Multiply $2$ by $- 1$ .

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Apply the constant rule.

$- x^{- 1} + C - (\frac{1}{3} x^{3} + C) - \frac{1}{3} x + C$

Step 10

Simplify.

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Step 10.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$- x^{- 1} + C - (\frac{x^{3}}{3} + C) - \frac{1}{3} x + C$

Step 10.2

Simplify.

$- \frac{1}{x} - \frac{x^{3}}{3} - \frac{1}{3} x + C$

Step 11

Reorder terms.

$- \frac{1}{x} - \frac{1}{3} x^{3} - \frac{1}{3} x + C$

Step 12

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

$F (x) =$ $- \frac{1}{x} - \frac{1}{3} x^{3} - \frac{1}{3} x + C$