7.2 definite integral — ગણિત ધોરણ 12 સાયન્સ — Question

$I = \int\limits_0^\pi  {\left[ {\cos \left( {\pi  - x} \right)} \right]dx} $

$ = \int\limits_0^\pi  {\left[ { - \cos x} \right]dx} \,\,\,\,\,......\left( 2 \right)$

On adding $(1)$ and $(2)$, we get 

$2I = \int\limits_0^\pi  {\left[ {\cos \left( {\pi  - x} \right)} \right]dx + } \int\limits_0^\pi  {\left[ { - \cos x} \right]dx} $

$2I = \int\limits_0^\pi  {\left[ {\cos x} \right] + } \left[ { - \cos x} \right]dx$

$2I = \int\limits_0^\pi  { - 1dx} $          ($\because $ $\left[ x \right] + \left[ { - x} \right] =  - 1$ if $x \notin {\rm Z}$)

$2I =  - \left. x \right|_0^\pi \,\, =  - \pi $

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