Calculus Examples

$f (t) = \frac{8}{1 + t^{2}}$

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

$F (t) = \int f (t) d t$

Set up the integral to solve.

$F (t) = \int \frac{8}{1 + t^{2}} d t$

Since $8$ is constant with respect to $t$ , move $8$ out of the integral.

$8 \int \frac{1}{1 + t^{2}} d t$

Rewrite $1$ as $1^{2}$ .

$8 \int \frac{1}{1^{2} + t^{2}} d t$

The integral of $\frac{1}{1^{2} + t^{2}}$ with respect to $t$ is $\arctan (t) + C$ .

$8 (\arctan (t) + C)$

Simplify.

$8 \arctan (t) + C$

The answer is the antiderivative of the function $f (t) = \frac{8}{1 + t^{2}}$ .

$F (t) =$ $8 \arctan (t) + C$