Calculus Examples

$\int \frac{1}{\sqrt{2 x - x^{2}}} d x$

Step 1

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

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

Step 2

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

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

Step 3

Reorder terms.

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