Answer
$$\frac{1}{\pi} \left(\frac{1}{5}\sin^5(\pi\theta)-\frac{1}{7}\sin^7 (\pi\theta) \right)+c $$
Work Step by Step
Given
$$ \int \cos ^{3}(\pi \theta) \sin ^{4}(\pi \theta) d \theta $$
Let $$u=\pi \theta\ \ \Rightarrow \ \ du =\pi d\theta$$
Then
\begin{align*}
\int \cos ^{3}(\pi \theta) \sin ^{4}(\pi \theta) d \theta&=\frac{1}{\pi} \int \cos ^{3} u \sin ^{4} u d u\\
&=\frac{1}{\pi} \int\left(1-\sin ^{2} u\right) \sin ^{4} u \cos u d u\\
&=\frac{1}{\pi} \int\left(\sin^4u-\sin ^{6} u\right) \cos u d u\\
&=\frac{1}{\pi} \left(\frac{1}{5}\sin^5u-\frac{1}{7}\sin^7 u \right)+c \\
&=\frac{1}{\pi} \left(\frac{1}{5}\sin^5(\pi\theta)-\frac{1}{7}\sin^7 (\pi\theta) \right)+c
\end{align*}
