Calculus Examples

$\int \frac{\cos (8 x)}{\sqrt{1 + \sin (8 x)}} d x$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{1 + \sin (8 x)}$ as ${(1 + \sin (8 x))}^{\frac{1}{2}}$ .

$\frac{d}{d x} [\int \frac{\cos (8 x)}{{(1 + \sin (8 x))}^{\frac{1}{2}}} d x]$

Step 2

$\int \frac{\cos (8 x)}{{(1 + \sin (8 x))}^{\frac{1}{2}}} d x$ is an antiderivative of $\frac{\cos (8 x)}{{(1 + \sin (8 x))}^{\frac{1}{2}}}$ , so by definition $\frac{d}{d x} [\int \frac{\cos (8 x)}{{(1 + \sin (8 x))}^{\frac{1}{2}}} d x]$ is $\frac{\cos (8 x)}{{(1 + \sin (8 x))}^{\frac{1}{2}}}$ .

$\frac{\cos (8 x)}{{(1 + \sin (8 x))}^{\frac{1}{2}}}$