An interesting limit

[Revolution]

Could someone tell me what

lim z->infi+ (Integral[cos^z(x),x ,0 ,pi/2])

is equal to?

 

[pka]

Could someone tell me what
lim z->infi+ (Integral[cos^z(x),x ,0 ,pi/2])
is equal to?

I am fairly sure that the limit is 0.
Here is why. For any \(\displaystyle t\in (0,\pi/2)\) the sequence \(\displaystyle \cos^n(t)\to 0~.\)
I have not worked out details, but I think that an application of the mean value theorem for integrals will complete the proof.