Easiest way to calculate this indefinite integral

Question

What's the easiest way to calculate the following indefinite integral:

$$ \int \frac{\cos(x)}{\sqrt{2\sin(x)+3}} \mathrm{d}x $$

Answer 1

Hint

$$\int\frac{f'(x)}{\sqrt{f(x)}}dx=2\sqrt{f(x)}+C$$

Answer 2

set $u=2\sin(x)+3$. then $du=2\cos(x)dx$. So it is $$ \int \frac{du}{2\sqrt{u}} $$

Answer 3

First, make the substitution $u = 2\sin(x) + 3$. Then, $du = 2\cos(x) \,dx$. Thus: $$\begin{align} \int \frac{\cos(x)}{\sqrt{2\sin(x) + 3}}dx &= \int \frac{du}{2\sqrt{u}}\\ &= \sqrt{u} + C\\ &= \ldots \end{align}$$

Answer 4

Here's a hint: $$ \int\frac{1}{\sqrt{2\sin x+3}}\Big( \cos x \, dx \Big) $$

If you don't know what that is hinting at, then that is what you need to learn about integration by substitution.