Calculus Examples

$h (x) = x^{3} - x$

The function $H (x)$ can be found by finding the indefinite integral of the derivative $h (x)$ .

$H (x) = \int h (x) d x$

Set up the integral to solve.

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

Split the single integral into multiple integrals.

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

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

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

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

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

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

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

Simplify.

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

The answer is the antiderivative of the function $h (x) = x^{3} - x$ .

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