If [x] denotes the greatest integer less than or equal to x, then the value of the integra…

Question

If $[x]$ denotes the greatest integer less than or equal to $x$, then the value of the integral $\displaystyle\int_{-\pi/2}^{\pi/2}[[x]-\sin x]\,dx$ is equal to :

Accepted Answer

$$I=\int_{-\pi/2}^{\pi/2}\left([x]+[-\sin x]\right)dx\ \ldots(1)$$

Using King's rule ($x\to-x$):
$$I=\int_{-\pi/2}^{\pi/2}\left([-x]+[\sin x]\right)dx\ \ldots(2)$$

Adding (1) and (2), and using $[x]+[-x]=-1$ (for non-integer $x$) and $[\sin x]+[-\sin x]=-1$:
$$2I=\int_{-\pi/2}^{\pi/2}(-2)\,dx=-2\pi\ \Rightarrow\ I=-\pi$$

Solutions · 1