For, x 2 ≠ n π + 1 , n ∈ N (the set of natural numbers), the integral ∫ x 2 sin x 2 - 1 - sin 2 x 2 - 1 2 sin x 2 - 1 + sin 2 x 2 - 1 d x , is equal to (where c is a constant of integration).

Step-by-step solution

⇒ I = ∫ x 2 s i n x 2 - 1 - s i n 2 x 2 - 1 2 s i n x 2 - 1 + s i n 2 x 2 - 1 d x Let, x 2 - 1 = θ ⇒ x d x = 1 2 d θ ⇒ I = 1 2 ∫ 2 s i n θ - s i n 2 θ 2 s i n θ + s i n 2 θ d θ = 1 2 ∫ 2 s i n θ - 2 s i n θ c o s θ 2 s i n θ + 2 s i n θ c o s θ d θ = 1 2 ∫ 1 - c o s θ 1 + c o s θ d θ = 1 2 ∫ t a n θ 2 d θ = 1 2 log e sec θ 2 1 2 + c , where c is the constant of integrat