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I am not even going to show my work because I have accomplished nothing attempting this problem. I have absolutely no idea what to do.

$$\int \frac{\cos^5 x}{ \sqrt {\sin x}}\, dx$$

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It's an application of the rule of thumb needed for this and several of your previous questions: If you have $\cos$ to an odd power upstairs in the integrand, try factoring out one $\cos$ term, write the other $\cos$ terms in terms of $\sin$, and let $u=\sin x$. –  David Mitra Jun 3 '12 at 15:56
    
@DavidMitra I have memorized that rule because it is in a table in my book. I know how to do simple problems like that but that was not my problem. –  user138246 Jun 3 '12 at 15:57
    
This silly system should tell me that someone else is editing the post at the same time. –  Gigili Jun 3 '12 at 15:58

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up vote 6 down vote accepted

Write $\cos^{5}(x) = (1-\sin^{2}{x})^{2} \cdot \cos{x}$ and then put $t =\sin{x}$.

So you will have \begin{align*} \int \frac{\cos^{5}(x)}{\sqrt{\sin{x}}} \ dx &= \int \frac{(1-\sin^{2}(x))^{2}}{\sqrt{\sin{x}}} \cdot \cos{x} \ dx \\\ &=\int \frac{(1-t^{2})^{2}}{t^{1/2}} \ dt \\\ &=\int \frac{1-2t^{2}+t^{4}}{t^{1/2}} \ dt \end{align*}

Don't forget to then substitute u= t^(1/2) so du = (1/2)u^(-1/2)

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How do I know that is the correct way of doing it? Out of the millions of ways to do this problem, why chose this way? I did it about 12 other ways and I never considered this way of doing it. Also I think your math is wrong, it seems to be missing the dt subsitution, shouldn't there be a 1/2? –  user138246 Jun 3 '12 at 15:56
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@Jordan you seem to have a rather negative attitude towards mathematics, and it is clear you don't like it and that you do it only because you have to. I'm sorry this is your situation as many of us get lost of fun, pleasure and intellectual challenge from mathematics, but if you have to do it then you might as well try a little harder and make a supreme effort to begin having a more positive attitude towards this stuff. That way you'll enjoy more (or will suffer less) and I'm sure you'll begin having more success. –  DonAntonio Jun 3 '12 at 16:14
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@Jordan Fair enough, but then don't get pesky at people doing their best to help you. And try to find some good time to invest in this, as it is my experience that people with an attitude simmilar to yours are in great danger of failing exams and stuff. –  DonAntonio Jun 3 '12 at 16:24
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@Chandrasekhar: there is a missing power in the second integral, i think. –  Chris's sis Jun 3 '12 at 16:34
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If that's true then you're doing it wrong. If you're looking for things to memorize then it would probably take that long, if you're looking for patterns and an understanding of what exactly these transformations achieve then it won't. Every time you solve a problem, or see an answer, look at what makes that answer work and try to come up with a new problem that can be solved that way. You need to see underneath the symbol pushing if you want any kind of payoff. Studying smarter is more important than studying harder. –  Robert Mastragostino Jun 3 '12 at 16:36

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