Evaluate the following integrals: ∫ e^2x ((1 - sin 2x)/(1 - cos 2x)) dx

Question

Evaluate the following integrals:

\(\int e^{2x}\left(\frac{1 - sin\,2x}{1 - cos\,2x}\right)dx\)

∫ e^2x ((1 - sin 2x)/(1 - cos 2x)) dx

Answer 1

\(\left(\frac{1 - sin\,2x}{1 - cos\,2x}\right)\)

((1 - sin 2x)/(1 - cos 2x))

Tip – If f₁(x) and f₂(x) are two functions , then an integral of the form ∫ f₁(x)f₂(x)dx can be INTEGRATED BY PARTS as

where f₁(x) and f₂(x) are the first and second functions respectively.

Taking f₁(x) = tan x and f₂(x) = e^2x in the second integral and keeping the first integral intact,

where c is the integrating constant.