If ∫ 0 π / 2 tan n x d x = k ∫ 0 π / 2 cot n x d x , then

Step-by-step solution

As, I = ∫ 0 π / 2 tan n x d x Apply the properties of definite integration: replace x → π 2 - t   ⇒   d x d t = - 1 , and upper limit x = π 2   ⇒   π 2 - t = π 2   ⇒   t = 0 Similarly, lower limit x = 0   ⇒   π 2 - t = 0   ⇒   t = π 2 I = ∫ 0 π / 2 tan n x d x = ∫ π 2 0 tan n π 2 - t - d t = - ∫ 0 π 2 tan n π 2 - t - d t = ∫ 0 π 2 tan n