Answer
$$\dfrac{1}{80 \pi}$$
Work Step by Step
$$\int_{R} dA= \int_0^{1/2} \int_0^{x^4} \cos (16 \pi x^5)dy dx\\ \int_0^{1/2} x^4 \cos (16 \pi x^5) dx$$
Suppose $16 \pi x^5 =t$ and $80 \pi x^4 dx=dt$
Now, $$\int_0^{1/2} x^4 \cos (16 \pi x^5) dx \\=\dfrac{1}{80} \int_0^{5 \pi/2} \cos (t) \space dt \\=\dfrac{1}{80 \pi}$$
