Calculus Examples

$\int (x + \sqrt{x}) d x$

Step 1

This integral could not be completed using u-substitution. Mathway will use another method.

Step 2

Remove parentheses.

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify.

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