Is there an easy way to calculate $\int_0^{2\pi} \sin^2x\ \cos^4x\ dx $? I want to calculate the integral
$$\int_0^{2\pi} \sin^2x\ \cos^4x\ dx  $$
by hand. The standard substitution $t=\tan(\frac{x}{2})$ is too difficult.
 Multiple integration by parts together with clever manipulations can give the result.
But it would be nice to have an easier method, perhaps one single substitution or some useful formulas for trigonometric powers.

Hint: $$e^{ix} = \cos x + i \sin x$$ and the binomial theorem.
Even further to simplify the algebra a bit initially you can use $$\sin^2x + \cos^2x = 1$$
