It's a problem I found nice and a little bit difficult so :
Let : $$\int_{0}^{\frac{\pi}{2}}\sqrt{\Big(\frac{\pi}{2}-x\Big)^{\alpha\cos^2(x)}-\Big(x\Big)^{\alpha\sin^2(x)}}dx=a+ib$$ Then prove that : $$a=b$$ Where $\alpha>0$ is a real number .
Clearly I don't know how to attack this problem .
I just know that :
$$\int_{0}^{\frac{\pi}{2}}\Big(\frac{\pi}{2}-x\Big)^{\alpha\cos^2(x)}-\Big(x\Big)^{\alpha\sin^2(x)}dx=0$$
After that I want to apply Cauchy-Schwarz's inequality (for integral) But I got nothing .
So have you a way to prove this ?
Thanks a lot for your time .