Question

can somone help me evaluate this integral

Answer 1

You have:
\[\int\limits_{\frac{\pi}{2}}^{0} \frac{1+\cos(2t)}{2}dt=\frac{1}{2} \int\limits_{\frac{\pi}{2}}^{0}1+\cos(2t)dt\]
For the second part use u=2t
du=2dt
(1/2)du=dt
Replacing this you have:
\[\frac{1}{2}\int\limits_{\frac{\pi}{2}}^{0}1dt+\frac{1}{2}(\frac{1}{2})\int\limits \cos(u)du=\frac{1}{2}t|_{\frac{\pi}{2}}^{0}+\frac{1}{4}\sin(u)\]
Replugging in your u you have:
\[[\frac{t}{2}+\frac{1}{4}\sin(2t)]_{\frac{\pi}{2}}^{0}=(0+0)-(\frac{\pi}{4}+\frac{1}{4}(0))=-\frac{\pi}{4}\]

Answer 2

thanks