Chapter 8: Techniques of Integration - Practice Exercises - Page 518: 43

$$4 \sqrt{2}\left|\sin \frac{t}{4}\right|+C $$

We integrate as follows: \begin{align*} \int \sqrt{1+\cos \left(\frac{t}{2}\right)} d t&=\int \sqrt{2}\left|\cos \frac{t}{4}\right| d t\\ &=4 \sqrt{2}\left|\sin \frac{t}{4}\right|+C \end{align*}