∫ e x 2 + sin ⁡ 2 x 1 + cos ⁡ 2 x   d x =

Step-by-step solution

Let, I = ∫ e x 2 + s i n ⁡ 2 x 1 + c o s ⁡ 2 x   d x We know that sin 2 x = 2 sin x cos x   &   cos 2 x = 2 cos 2 x - 1 Using the above formulas we can write I = ∫ e x 2 + 2 sin x cos x 2 cos 2 x d x I = ∫ e x 2 1 + sin x cos x 2 cos 2 x d x I = ∫ e x 1 cos 2 x + sin x cos x cos 2 x d x I = ∫ e x ( tan x + sec 2 x ) d x We know that ∫ e x f x + f ' x dx = e x · f + C Using the above formula we can write I = e x tan x + C