Calculus Examples

$x^{2} + \sqrt{x}$

Step 1

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

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

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.

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

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

Step 6

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$\frac{1}{3} x^{3} + C + \int x^{\frac{1}{2}} d x$

Step 7

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

$\frac{1}{3} x^{3} + C + \frac{2}{3} x^{\frac{3}{2}} + C$

Step 8

Simplify.

$\frac{1}{3} x^{3} + \frac{2}{3} x^{\frac{3}{2}} + C$

Step 9

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

$F (x) =$ $\frac{1}{3} x^{3} + \frac{2}{3} x^{\frac{3}{2}} + C$