Answer
$1638.4$
Work Step by Step
$I=\int_{-\pi/2}^{\pi/2} 4 cos(t) (4 sint)^4 (4 dt)=(4^{6})\int_{-\pi/2}^{\pi/2} cos(t)sin^4(t) $
Plug $\sin t=a \implies \cos t dt =da$ and $ \sin(-\pi/2) = -1$;$ \sin(\pi/2) = 1$
$=4096 \int_{-\pi/2}^{\pi/2} 1024 [\cos (t) \times \sin^4(t) dt]$
$ = 4096 \int_{-1}^{1} a^4 da$
$= 4096 [(1/5) a^5]^{1}_{-1}$
$=\dfrac{4096} ( \dfrac {2}{5})$
$\approx 1638.4$
