The value of the integral Ι = ∫ d x 1 + s i n x , ∀ x ∈ 0 , π 2 is equal to k ln ⁡ t a n π 4 + x 8 + c , then the value of k 2 is equal to (where, c is the constant of integration)

Step-by-step solution

As, 1 + sin ⁡ x = s i n 2 x 2 + c o s 2 x 2 + 2 s i n x 2 c o s x 2 , = s i n x 2 + c o s x 2 2 Ι = ∫ d x s i n x 2 + c o s x 2 = 1 2 ∫ d x s i n π 4 + x 2 = 1 2 ∫ cosec π 4 + x 2 d x = 1 2 2 ln ⁡ t a n π 8 + x 4 + c = 2 ln ⁡ t a n π 8 + x 4 + c ⇒ k = 2 Hence, 2 k = 2