Calculus Examples

$\int \frac{\sqrt{x^{2} - 4}}{x} d x$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} - 4}$ as ${(x^{2} - 4)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [\int \frac{{(x^{2} - 4)}^{\frac{1}{2}}}{x} d x]$

Step 2

$\int \frac{{(x^{2} - 4)}^{\frac{1}{2}}}{x} d x$ is an antiderivative of $\frac{{(x^{2} - 4)}^{\frac{1}{2}}}{x}$ , so by definition $\frac{d}{d x} [\int \frac{{(x^{2} - 4)}^{\frac{1}{2}}}{x} d x]$ is $\frac{{(x^{2} - 4)}^{\frac{1}{2}}}{x}$ .

$\frac{{(x^{2} - 4)}^{\frac{1}{2}}}{x}$