Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Reorder terms.

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