Answer
$4 \sqrt {2} |\sin(t/4)|+c$
Work Step by Step
Consider the integral $\int \sqrt {1+\cos (t/2)}dt$
The given integral can be re-written as:
$\int \sqrt {1+(2\cos^2 (t/4)-1)}dt=\sqrt 2 \int \cos (t/4) dt$
This implies that $4 \sqrt {2 \sin^2(t/4)}+c=4 \sqrt {2} |\sin(t/4)|+c$