The integral Ι = ∫ e x 1 + sin ⁡ x 1 + cos ⁡ x d x = e x f x + C (where, C is the constant of integration). Then, the range of y = f x (for all x in the domain of f x ) is

Step-by-step solution

The given integral Ι = ∫ e x 1 + 2 s i n x 2 c o s x 2 2 c o s 2 x 2 d x = ∫ e x 1 2 s e c 2 x 2 + t a n x 2 d x = e x ⋅ tan x 2 + C   ∵   ∫ e x f x + f ' x d x = e x f x + C Thus, f x = t a n x 2 whose range is - ∞ , ∞