(a) Evaluate: ∫_0^π / 2 sin 2 x cos 3 x d x

$$\begin{aligned} & I=\int_{0}^{\frac{\pi}{2}} \sin 2 x \cos 3 x d x \\ & =\frac{1}{2} \int_{0}^{\frac{\pi}{2}}(\sin 5 x-\sin x) d x \\ & =\frac{1}{2}\left[-\frac{1}{5} \cos 5 x+\cos x\right]_{0}^{\frac{\pi}{2}} \\ & =-\frac{2}{5} \end{aligned} $$ $$\begin{aligned} & F(x)=\int \frac{1}{\sqrt{2 x-x^{2}}} d x \\ & =\int \frac{1}{\sqrt{1-(x-1)^{2}}} d x \\ & =\sin ^{-1}(x-1)+c \\ & \text { when } x=1, y=0 \text { gives } c=0 \\ & \therefore F(x)=\sin ^{-1}(x-1) \end{aligned} $$