Evaluate $\int \sqrt{1 + \sin(x/2)} dx$, assuming $x \in (0, \pi)$.

Question

Professor PC Thomas Classes · Answer

Using trigonometric identities, $1 + \sin(x/2) = (\sin(x/4) + \cos(x/4))^2$. Since $x \in (0, \pi)$, $x/4 \in (0, \pi/4)$, so both sine and cosine are positive. The integral is $\int (\sin(x/4) + \cos(x/4)) dx = -4\cos(x/4) + 4\sin(x/4) + C = 4\sin(x/4) - 4\cos(x/4) + C$.