$$\int \cos^4 x\,\mathrm dx = \int (1 - \sin^2x)^2\,\mathrm dx.$$
I tried using $\cos^2(x) + \sin^2(x)$ = 1. This was on the integration by parts section of my textbook.
The integral I came out with is given me a hard time. I double checked my work and I feel I am on the right track but I feel stuck. I tried u substitution but there is no $\cos x$ to leverage with.