The value of the integral ∫ 1 / 2 2 tan - 1 x x d x is equal to

Step-by-step solution

Let, I = ∫ 1 2 2 tan - 1 x x d x   . . . . . . . . 1 Now, let x = 1 t ⇒ d x = - 1 t 2 d t So, I = - ∫ 2 1 2 tan - 1 1 t 1 t × 1 t 2 d t ⇒ I = - ∫ 2 1 2 tan - 1 1 t × 1 t d t ⇒ I = ∫ 1 2 2 cot - 1 t × 1 t d t ⇒ I = ∫ 1 2 2 cot - 1 x × 1 x d x   . . . . . . . 2 Now adding 1   &   2 , we get, 2 I = ∫ 1 2 2 1 x cot - 1 x + tan - 1 x d x ⇒ 2 I = ∫ 1 2 2 1 x × π 2 d x ⇒ 2 I = π 2 ln x 1