$\int\limits_{0}^{\pi/4}\cos^{3}2x dx$
Put $2x=\theta$
$2x=d \theta$
$dx =\large\frac{1}{2}$$ d \theta$
When $x=0=> \theta=0$
When $x=\large\frac{\pi}{2}$$=> \theta=2 \times \large\frac{\pi}{4}=\frac{\pi}{2}$
$\int\limits_{0}^{\pi/4}\cos^{3}2x dx= \int\limits_{0}^{\pi/2}\cos^{3} \theta \large\frac{1}{2} d \theta$
$\qquad= \large\frac{1}{2} \int\limits_{0}^{\pi/2}$$ \cos^{3} \theta d \theta$
$\qquad= \large\frac{1}{2} . \frac{3-1}{3} .1$
$\qquad= \large\frac{1}{2} . \frac{2}{3} = \frac{1}{3}$
Hence 2 is the correct answer.