Trig Integration, how did they get this answer? 
Here is the full question for context:



And here is the full answer given:


I don't understand how they go from the first part in the red box to the next, I don't have a clue.

It's a $u$ substitution where they take $u = \sin(x)$. If $u = \sin(x)$, then $\mathrm{d}u = \cos(x) \;\mathrm{d}x$. So we have
$$\begin{align}
  \int {\sin^{2n}(x)\cos(x)} \;\mathrm{d}x \;\;&=\;\; 
  \int {u^{2n}} \;\mathrm{d}u \\\;\;&=\;\;
  \frac{u^{2n+1}}{2n+1} \;\;=\;\;
  \frac{\sin^{2n+1}(x)}{2n+1}
\end{align}$$
