Calculus Examples

$\int (x^{4} + \sqrt{x^{3}}) d x$

Step 1

Remove parentheses.

$\int x^{4} + \sqrt{x^{3}} d x$

Step 2

Split the single integral into multiple integrals.

$\int x^{4} d x + \int \sqrt{x^{3}} d x$

Step 3

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$\frac{1}{5} x^{5} + C + \int \sqrt{x^{3}} d x$

Step 4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{3}}$ as $x^{\frac{3}{2}}$ .

$\frac{1}{5} x^{5} + C + \int x^{\frac{3}{2}} d x$

Step 5

By the Power Rule, the integral of $x^{\frac{3}{2}}$ with respect to $x$ is $\frac{2}{5} x^{\frac{5}{2}}$ .

$\frac{1}{5} x^{5} + C + \frac{2}{5} x^{\frac{5}{2}} + C$

Step 6

Simplify.

$\frac{1}{5} x^{5} + \frac{2}{5} x^{\frac{5}{2}} + C$