Question

The value of definite integral $\int\limits_0^{\frac{\pi}{2}} \sin \left(\sin ^{-1}[x]\right) d x$ is equal to
[Note: $[ k ]$ denotes the largest integer less than or equal to $k$ ]

• A. $\pi-1$
• B. $\frac{\pi}{2}$
• C. $\frac{\pi-2}{2}$
• D. $\frac{\pi}{4}$

Answer 1

Answer: (C)

Let $f(x)=\sin \left(\sin ^{-1}[x]\right)=\begin{cases}0, & 0 \leq x<1 \\ 1, & 1 \leq x<\frac{\pi}{2}\end{cases}$
$\therefore I =\int\limits_0^1 0 dx +\int\limits_1^{\frac{\pi}{2}} 1 dx =0+\left(\frac{\pi}{2}-1\right)=\frac{\pi-2}{2}$