Calculus Examples

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

Step 1

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

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

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.

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$\ln (| x |) + C + \int - x^{2} d x$

Step 6

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

$\ln (| x |) + C - \int x^{2} d x$

Step 7

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

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

Step 8

Simplify.

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

Step 9

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

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