Answer
$\dfrac{1}{80 \pi}$
Work Step by Step
We reverse the order of integration to get:
$I= \int_0^{1/2} \int_0^{x^4} \cos (16 \pi x^5)dy dx$
or, $= \int_0^{1/2} x^4 \cos (16 \pi x^5) dx$
Set $a = 16 \pi x^5 \implies da=80 \pi x^4 dx$
Now, $I=\dfrac{1}{80} \int_0^{5 \pi/2} \cos a da=\dfrac{1}{80 \pi}$
