Calculus Examples

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

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

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

Set up the integral to solve.

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

Split the single integral into multiple integrals.

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

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

$\frac{1}{3} x^{3} + C + \int d x$

Apply the constant rule.

$\frac{1}{3} x^{3} + C + x + C$

Simplify.

$\frac{1}{3} x^{3} + x + C$

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

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