# Easiest way to calculate this indefinite integral

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

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

• Notice that the numerator is (up to a constant factor) the derivative of the radicand. – Daniel Fischer Jan 4 '14 at 21:56
• @DanielFischer : I phrased that same thought in a rather different way in my posted answer. But I wonder if it's reasonable to expect a lay reader to understand technical terms like "up to". – Michael Hardy Jan 4 '14 at 22:31

## 4 Answers

Hint

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

• Thanks a lot for this interesting approach, I will accept your answer in next 5 minutes. – NullPointer Jan 4 '14 at 22:04
• I think you're missing an integral sign there. – David Zhang Jan 5 '14 at 1:09
• Of course, hehe. Thanks, @DavidZhang – DonAntonio Jan 5 '14 at 4:07

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

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.

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}