Answer
$-\frac{1}{2}-\frac{\pi}{4}$
Work Step by Step
Substitute $a=\theta^2$. Then $da=2\theta\thinspace d\theta$.
When substituting the limits into $\theta$, they become $\frac{\pi}{2}$ and $\pi$.
Our integral is now $\int_{\frac{\pi}{2}}^{\pi}\frac{1}{2}acos(a)\thinspace da=\frac{1}{2}\int_{\frac{\pi}{2}}^{\pi}acos(a)\thinspace da$.
We'll have $u=a$ and $dv=cos(a)da$.
$u=a$
$du=da$
$dv=cos(a)da$
$v=sin(a)$
$\frac{1}{2}\left[asin(a)-\int sin(a)da\right]_{\frac{\pi}{2}}^{\pi}$
$=\frac{1}{2}\left[asin(a)+cos(a)\right]_{\frac{\pi}{2}}^{\pi}$
$=\frac{1}{2}\left[\pi sin(\pi)+cos(\pi)-(\frac{\pi}{2} sin(\frac{\pi}{2})+cos(\frac{\pi}{2}))\right]$
$=\frac{1}{2}\left[\pi (0)+(-1)-(\frac{\pi}{2}(1)+0)\right]$
$=-\frac{1}{2}-\frac{\pi}{4}$
