Answer
$$
\int_{0}^{\pi} \cos ^{6} \theta d \theta =\frac{5 \pi}{16}
$$
Work Step by Step
$$
\int_{0}^{\pi} \cos ^{6} \theta d \theta
$$
If we look at the Table of Integrals , we see that the closest entry is number $74$ with $n=6 $, for the second integral we use Formula $74$ with $n=4,$ and for the last integral we use Formula $64$ :
$$
\begin{aligned}
\int_{0}^{\pi} \cos ^{6} \theta d \theta & \stackrel{74}{=}\left[\frac{1}{6} \cos ^{5} \theta \sin \theta\right]_{0}^{\pi}+\frac{5}{6} \int_{0}^{\pi} \cos ^{4} \theta d \theta \\
&\stackrel{74}{=} 0+\frac{5}{6}\left\{\left[\frac{1}{4} \cos ^{3} \theta \sin \theta\right]_{0}^{\pi}+\frac{3}{4} \int_{0}^{\pi} \cos ^{2} \theta d \theta\right\} \\
&\stackrel{64}{=} \frac{5}{6}\left\{0+\frac{3}{4}\left[\frac{1}{2} \theta+\frac{1}{4} \sin 2 \theta\right]_{0}^{\pi}\right\}=\frac{5}{6} \cdot \frac{3}{4} \cdot \frac{\pi}{2}\\
&=\frac{5 \pi}{16}
\end{aligned}
$$
