If [,] denotes the greatest integer function, then the integral _0^ [ x],dx is equal to:

Question

If $[\,]$ denotes the greatest integer function, then the integral $\displaystyle\int_0^\pi [\cos x]\,dx$ is equal to:

Accepted Answer

Let $I = \int_0^\pi [\cos x]\,dx$ (1)

$I = \int_0^\pi [\cos(\pi-x)]\,dx = \int_0^\pi [-\cos x]\,dx$ (2)

Adding (1) and (2): $2I = \int_0^\pi \left([\cos x]+[-\cos x]ight)dx = \int_0^\pi (-1)\,dx$ (since $[y]+[-y]=-1$ for $y 
otin \mathbb{Z}$)

$2I = -\pi \Rightarrow I = -\dfrac{\pi}{2}$

Solutions · 1