If ∫ 0 a a - x x d x = K 2 , then K = ….

Step-by-step solution

We have, ∫ 0 a a - x x d x = k 2 Let l = ∫ 0 a a - x x d x Put x = a s i n 2 θ ⇒ d x = a 2 s i n θ c o s θ d θ When, x = 0 , θ = 0 a n d x = a , θ = π 2 ∴ ∫ 0 π / 2 a - a s i n 2 θ a s i n 2 θ 2 a s i n θ c o s θ d θ = 2 a ∫ 0 π / 2 ( c o t θ ) s i n θ c o s θ d θ = 2 a ∫ 0 π / 2 c o s 2 θ d θ = 2 a ∫ 0 π / 2 1 + c o s 2 θ 2 d θ = a ∫ 0 π / 2 1 + c o s 2 θ d θ = a θ + s i n 2 θ 2 0 π / 2 = a π 2 + s i n π 2 = π a 2 ∴ k = π a