Calculus Examples

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

Step 1

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

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

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.

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

Step 4

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

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

Step 5

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

$\arctan (x) + C$

Step 6

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

$F (x) =$ $\arctan (x) + C$