Answer
$$
\int_{0}^{\pi / 2} f(2 \sin \theta) \cos \theta d \theta=3
$$
Work Step by Step
$$
\int_{0}^{\pi / 2} f(2 \sin \theta) \cos \theta d \theta
$$
Let $u=2 \sin \theta $. Then $du=2\cos \theta d \theta. $
When $\theta = 0, u = 0 $; when $\theta = \pi / 2, u = 2$.
Thus, the Substitution Rule gives
$$
\begin{aligned}
\int_{0}^{\pi / 2} f(2 \sin \theta) \cos \theta d \theta &=\int_{0}^{2} f(u)\left(\frac{1}{2} d u\right) \\
&=\frac{1}{2} \int_{0}^{2} f(u) d u \\
&=\frac{1}{2} \int_{0}^{2} f(x) d x \\
&=\frac{1}{2}(6) \\
&=3
\end{aligned}$$
