Calculus Examples

$\int \sqrt{x^{3}} + \sqrt[3]{x^{2}} d x$

Step 1

Split the single integral into multiple integrals.

$\int \sqrt{x^{3}} d x + \int \sqrt[3]{x^{2}} d x$

Step 2

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

$\int x^{\frac{3}{2}} d x + \int \sqrt[3]{x^{2}} d x$

Step 3

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

$\frac{2}{5} x^{\frac{5}{2}} + C + \int \sqrt[3]{x^{2}} d x$

Step 4

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

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

Step 5

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

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

Step 6

Simplify.

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