If ∫ 0 π / 3 tan ⁡ θ 2 k   sec ⁡ θ d θ = 1 - 1 2 , ( k > 0 ) , then the value of k is

Step-by-step solution

∫ 0 π / 3 tan ⁡ x 2 k sec ⁡ x d x = 1 2 k ∫ 0 π / 3 sin ⁡ x cos ⁡ x d x Put t = cos ⁡ x d t = - sin ⁡ x d x , also x = 0 , t = 1 ; x = π 3 , t = 1 2 ∴ 1 2 k ∫ 1 1 / 2 - d t t = 1 2 k ∫ 1 / 2 1 t - 1 2 d t = 1 2 k × 2 t 1 2 1 / 2 1 = 2 k 1 - 1 2 = 2 - 1 k Given 2 - 1 k = 1 - 1 2 = 2 - 1 2 ∴ k = 2