Calculus Examples

$\int \sqrt[5]{x} + \sqrt[6]{x} d x$

Step 1

Split the single integral into multiple integrals.

$\int \sqrt[5]{x} d x + \int \sqrt[6]{x} d x$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{x}$ as $x^{\frac{1}{5}}$ .

$\int x^{\frac{1}{5}} d x + \int \sqrt[6]{x} d x$

Step 3

By the Power Rule, the integral of $x^{\frac{1}{5}}$ with respect to $x$ is $\frac{5}{6} x^{\frac{6}{5}}$ .

$\frac{5}{6} x^{\frac{6}{5}} + C + \int \sqrt[6]{x} d x$

Step 4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[6]{x}$ as $x^{\frac{1}{6}}$ .

$\frac{5}{6} x^{\frac{6}{5}} + C + \int x^{\frac{1}{6}} d x$

Step 5

By the Power Rule, the integral of $x^{\frac{1}{6}}$ with respect to $x$ is $\frac{6}{7} x^{\frac{7}{6}}$ .

$\frac{5}{6} x^{\frac{6}{5}} + C + \frac{6}{7} x^{\frac{7}{6}} + C$

Step 6

Simplify.

$\frac{5}{6} x^{\frac{6}{5}} + \frac{6}{7} x^{\frac{7}{6}} + C$