Calculus Examples

$g (t) = t^{3} + t^{2}$

The function $G (t)$ can be found by finding the indefinite integral of the derivative $g (t)$ .

$G (t) = \int g (t) d t$

Set up the integral to solve.

$G (t) = \int t^{3} + t^{2} d t$

Split the single integral into multiple integrals.

$\int t^{3} d t + \int t^{2} d t$

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

$\frac{1}{4} t^{4} + C + \int t^{2} d t$

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

$\frac{1}{4} t^{4} + C + \frac{1}{3} t^{3} + C$

Simplify.

$\frac{1}{4} t^{4} + \frac{1}{3} t^{3} + C$

The answer is the antiderivative of the function $g (t) = t^{3} + t^{2}$ .

$G (t) =$ $\frac{1}{4} t^{4} + \frac{1}{3} t^{3} + C$