The value of the integral ∫ 0 π e cos ⁡ x sin ⁡ x 1 + e cot ⁡ x d x is equal to

Step-by-step solution

Let, Ι = ∫ 0 π e cos ⁡ x sin ⁡ x 1 + e cot ⁡ x d x … 1 Ι = ∫ 0 π e cos ⁡ x sin ⁡ x 1 + e - cot ⁡ x d x … 2 A p p l y i n g   a + b - x   property 1 + 2 , we get, 2 I = ∫ 0 π e cos ⁡ x sin ⁡ x d x Putting cos ⁡ x = t and - sin ⁡ x d x = d t I = 1 2 ∫ 1 - 1 - e t d t I = 1 2 ∫ - 1 1 ( e t ) d t I = ∫ 0 1 e t d t = e - 1 A s   e t   i s   e v e n I = e - 1