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I would like to know if there is some way to calculate exactly

$$\int_{0}^{\pi/2} \frac{\cos^3 x}{\sqrt {\sin x}(1+\sin^3 x)}\,dx$$

The numerical value is 1,52069

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    Mathematica gives the exact answer as $\frac{\ln(7+4\sqrt{3})}{\sqrt{3}}$.2011-12-25
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    Try substituting $y = \sqrt{\sin(x)}$ which changes the integral to $$\int_0^1\frac{1-y^4}{1+y^6}\mathrm dx.$$ Does that help any?2011-12-25
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    Since $\cos(x)$ only appears at an odd power, the standard substitution $u=\sin(x)$ works...2011-12-25
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    @DilipSarwate I think you need to change the $dx$ to $dy$ and the integral in question is twice your integral.2011-12-25
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    @KannappanSampath Yes, you are correct. Unfortunately, comments cannot be edited except right away...2011-12-25

4 Answers 4