Question

Evaluate: $\int \left[\sin x\right]dx$ for $x\in (0,\frac{\pi }{2})$, where [.] represents greatest integer function.

• A. $0$
• B. $\cos x+c$, $c$ is a constant of integration
• C. $c$, $c$ is a constant of integration
• D. None of the above

Answer 1

Answer: (C)

$c$, $c$ is a constant of integration

We know that the range of $\sin x$ in $(0,\frac{\pi }{2})$ is in $(0,1)$.

$\therefore \left[\sin x\right]=0$ for $x\in (0,\frac{\pi }{2})$

Hence,

$I=\int 0dx=0+c=c$ {where $c$ is the constant of integration}