Answer
$-2$
Work Step by Step
Our aim is to solve the integral $ \int_0^{\pi} \cos(x+\dfrac{\pi}{2}) \ dx $
Let us consider that $a =x+\dfrac{\pi}{2}$ and $dx=da$
Now, $\int \cos(x+\dfrac{\pi}{2}) \ dx = \int \cos a \ da\\=\sin a +C\\=\sin (x+\dfrac{\pi}{2}) +C$
Now, $ \int_0^{\pi} \cos(x+\dfrac{\pi}{2}) \ dx=[\sin (x+\dfrac{\pi}{2})]_0^{\pi} $
or, $=\sin (\pi+\dfrac{\pi}{2})-\sin (0+\dfrac{\pi}{2})$
or, $ =-1-1$
or, $=-2$
