Evaluate the following integrals:

We know that

Putting in the original equation

Using BY PART METHOD. Using the superiority list as ILATE (Inverse Logarithm Algebra Trigonometric Exponential). Taking the first function to the one which comes first in the list.

Here cos6x and cos2x is first function and e ^{- x} as the second function.

Solving both parts individually

Solving the second part,

Putting in the obtained equation