This will be about integrating product of trigonometric functions with no powers involved for example,

$\int sin6xcos21x dx$ or $\int cosxcos5x dx$.

The simple trick here is to use the product to sum to formulas that can be found here.
For convenience, I’ll insert them here.

So very conveniently,

$\int cosxcos5x dx=\int \frac{1}{2}(cos4x+cosx) dx = \frac{1}{2}(\frac{sin4x}{4}+sinx)$.
Now you see the usefulness of the product to sum formula! Better learn how to find them!

For trigonometric functions with powers involved, you can refer to part 1 to part 5.