The integral shown is:

$I = \int \sin^4 x \cos^2 x \, dx$

To solve it, the problem mentions using double angle identities and trigonometric identities. The identity mentioned is:

$\cos u \cos v = \frac{1}{2} [\cos(u - v) + \cos(u + v)]$

Let's start by rewriting the powers of sine and cosine in terms of trigonometric identities. Specifically, we can express $\sin^4 x$ and $\cos^2 x$ using identities to simplify the expression and integrate.

We'll follow the steps of applying trigonometric transformations, expanding terms, and then integrating. Let me proceed with the calculation.

Do you want me to show you the detailed steps or provide the answer directly?