Any thoughts on this integral? $\int \cos^2(x)\cdot\sin^4(x)dx$
I tried the usual trigonometric identities but they don't seem helpful

Write $c$ for $\cos x$ and $s$ for $\sin x$. Then the integrand is 
\begin{align}
c^2 s^4 
&= c^2 s^2 (s^2) \\
&= \frac{1}{4} (2cs)^2 (1 - c^2) \\
&= \frac{1}{2} \frac{1}{4} (2cs)^2 (2 - 2c^2)  \\
&= \frac{1}{2} \left( \frac{1}{4} (2cs)^2 (1 - 2c^2)  \right) 
 + \frac{1}{2} \left( \frac{1}{4} (2cs)^2   \right) \\
\end{align}
Now $2cs = \sin 2x$, and $1 - 2c^2 = -\cos 2x$, so from here things should be relatively simple. 
