Answer
$$\frac{1}{13} \sin \left(\frac{13 \theta}{2}\right)+\frac{1}{15} \sin \left(\frac{15 \theta}{2}\right) +C$$
Work Step by Step
Use the formula
$$\int \cos a x \cos b x d x=\frac{\sin (a-b) x}{2(a-b)}+\frac{\sin (a+b) x}{2(a+b)}+C, \quad a^{2} \neq b^{2}$$
with $ a=\dfrac{1}{3},\ \ \ b =\dfrac{1}{4}$, then we get
\begin{align*}
\int \cos \frac{\theta}{2} \cos 7 \theta d \theta=\frac{1}{13} \sin \left(\frac{13 \theta}{2}\right)+\frac{1}{15} \sin \left(\frac{15 \theta}{2}\right)+ c
\end{align*}
