Question

Evaluate the following integrals:

∫ e^-x cos 2x cos 4x dx

Answer 1

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 cos 6x and cos 2x is first function and e^{- x} as the second function.

Solving both parts individually

Solving the second part,

Putting in the obtained equation

Answer 2

Using the formula cos A cos B = 1/2 [cos (A + B) + cos (A – B)] We get cos 4x cos 2x = 1/2 [cos (4x + 2x) + cos (4x – 2x)] = 1/2 [cos 6x + cos 2x] By applying in the original equation ∫ e–x cos 2x cos 4x dx = ∫ e–x (1/2[cos 6x + cos 2x]) = 1/2 [∫e–x cos 6x dx + ∫e–x cos 2x dx] Taking first function as cos 6x and cos 2x and second function as e–x Now by solving both parts separately