Calculus Examples

$\int 5 \cos (3 x) \sqrt{\sin (3 x + 4)} d x$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\sin (3 x + 4)}$ as ${\sin (3 x + 4)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [\int 5 \cos (3 x) {\sin (3 x + 4)}^{\frac{1}{2}} d x]$

Step 2

$\int 5 \cos (3 x) {\sin (3 x + 4)}^{\frac{1}{2}} d x$ is an antiderivative of $5 \cos (3 x) {\sin (3 x + 4)}^{\frac{1}{2}}$ , so by definition $\frac{d}{d x} [\int 5 \cos (3 x) {\sin (3 x + 4)}^{\frac{1}{2}} d x]$ is $5 \cos (3 x) {\sin (3 x + 4)}^{\frac{1}{2}}$ .

$5 \cos (3 x) {\sin (3 x + 4)}^{\frac{1}{2}}$

Step 3

Reorder the factors of $5 \cos (3 x) {\sin (3 x + 4)}^{\frac{1}{2}}$ .

$5 {\sin (3 x + 4)}^{\frac{1}{2}} \cos (3 x)$