(a)[4]
Evaluate $\displaystyle \int \left( \frac{8}{4x + 1} + \frac{8}{\cos^2(4x + 1)} \right)\,dx.$
(b)[6]
It is stated that $\int_{0}^{\frac{1}{2}\pi} \left(3 + 4\cos^2 \frac{1}{2}x + k\sin 2x \right) dx = 10$. Determine the exact value of the constant $k$.