Answer
$$\frac{\sqrt{2}+7}{45}$$
Work Step by Step
Given
$$ \int_{0}^{\pi/4}\sin 7 x\cos 2xdx$$
Use
$$ \sin (a x) \cos (b x)=\frac{1}{2} \sin ((a-b) x)+\frac{1}{2} \sin ((a+b) x) $$
Then
\begin{align*}
\int_{0}^{\pi/4}\sin 7 x\cos 2xdx&=\frac{1}{2}\int_{0}^{\pi/4}(\sin (5x)+ \sin 9x)dx\\
&=\frac{ -1}{2}\left(\frac{1}{5}\cos5 x+\frac{1}{9}\cos9 x\right)\bigg|_{0}^{\pi/4} \\
&=\frac{\sqrt{2}+7}{45}
\end{align*}
