(sin8x)13(cosx)12\frac{(\sin 8x)^{\frac{1}{3}}}{(\cos x)^{\frac{1}{2}}} Gamma & Beta Function

Compute

 ∫π/20(sin8x)1/3(cosx)1/2dx.\ \int_0^{\pi/2}\dfrac{(\sin 8x )^{1/3}}{(\cos x)^{1/2}} dx .

using Gamma & Beta Function properties

This question make me suffer a lot.

I have used everything I am capable for and I failed
I evaluated this integral using Matlab , and the result was an imaginary number.
If anyone has any idea about this question please share it with me.

=================

  

 

There is indeed a big problem of definition because sin(8x)\sin(8x) is frequently negative on the interval of integration. Thus its power 1/31/3 will generate complex numbers. I doubt that this integral is amenable to a Beta integral…
– JeanMarie
2 days ago

  

 

so probably there is a mistake n the question @JeanMarie
– Feras Mohammed
2 days ago

  

 

@JeanMarie: I assume they use 3√−1=−1\sqrt[3]{-1}=-1.
– robjohn♦
2 days ago

=================

=================