Calculus Examples

$\int \sqrt[11]{x} + \sqrt[12]{x} d x$

Step 1

Split the single integral into multiple integrals.

$\int \sqrt[11]{x} d x + \int \sqrt[12]{x} d x$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[11]{x}$ as $x^{\frac{1}{11}}$ .

$\int x^{\frac{1}{11}} d x + \int \sqrt[12]{x} d x$

Step 3

By the Power Rule, the integral of $x^{\frac{1}{11}}$ with respect to $x$ is $\frac{11}{12} x^{\frac{12}{11}}$ .

$\frac{11}{12} x^{\frac{12}{11}} + C + \int \sqrt[12]{x} d x$

Step 4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[12]{x}$ as $x^{\frac{1}{12}}$ .

$\frac{11}{12} x^{\frac{12}{11}} + C + \int x^{\frac{1}{12}} d x$

Step 5

By the Power Rule, the integral of $x^{\frac{1}{12}}$ with respect to $x$ is $\frac{12}{13} x^{\frac{13}{12}}$ .

$\frac{11}{12} x^{\frac{12}{11}} + C + \frac{12}{13} x^{\frac{13}{12}} + C$

Step 6

Simplify.

$\frac{11}{12} x^{\frac{12}{11}} + \frac{12}{13} x^{\frac{13}{12}} + C$