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

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

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Notice that the numerator is (up to a constant factor) the derivative of the radicand. –  Daniel Fischer Jan 4 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 at 22:31
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4 Answers 4

up vote 6 down vote accepted

Hint

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

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Thanks a lot for this interesting approach, I will accept your answer in next 5 minutes. –  NullPointer Jan 4 at 22:04
    
I think you're missing an integral sign there. –  David Zhang Jan 5 at 1:09
    
Of course, hehe. Thanks, @DavidZhang –  DonAntonio Jan 5 at 4:07
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set $u=2\sin(x)+3$. then $du=2\cos(x)dx$. So it is $$ \int \frac{du}{2\sqrt{u}} $$

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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.

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

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