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I am struggling with a particular definite integral. Here it goes:

$$ \int_0^{\pi/2} \frac{\sin^{1/3}(x) \, \mathrm{d}x}{\sin^{1/3}(x) + \cos^{1/3}(x)} $$

I have tried several substitutions but haven't succeeded. Mathematica says that the result is $\pi/4$, and it even gives an analytic solution for the indefinite integral. However, it doesn't really help me to see how it is worked out.

Would you please have a look and hint me in the direction/solve it? Thanks a lot.

SSF

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1 Answer 1

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It is just $\frac{\pi}{4}$. Apply the substitution $x\mapsto\frac{\pi}{2}-x$ and add the resulting integrals to get $\int_{0}^{\pi/2}1\,dx.$

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  • $\begingroup$ That's really clever! Thanks! $\endgroup$
    – SSF
    Commented Nov 9, 2015 at 21:15
  • $\begingroup$ Very clever indeed. $\endgroup$
    – hbp
    Commented Nov 12, 2015 at 3:01

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