Let ∫ sin ⁡ 2 x ln ⁡ ( cos ⁡ x ) d x = ⁡ f ( x ) cos 2 ⁡ x + C , (where, C is the constant of integration) and f 0 = 1 2 . If f π 3 is equal to 1 a + ln ⁡ b , then the value of a + b is

Step-by-step solution

Given integral is I = ∫ 2 sin ⁡ x cos ⁡ x ln ⁡ ⁡ ( cos ⁡ x ) d x Let, cos ⁡ x = t ⇒ - sin ⁡ x d x = d t ⇒ I = - 2 ∫ t ln ⁡ t d t Using Integration by parts I = - 2 t 2 2 ln ⁡ t - ∫ t 2 2 1 t d t I = - t 2 ln ⁡ t + t 2 2 + C I = cos 2 ⁡ x 1 2 - ln ⁡ cos ⁡ x + C f x = 1 2 - ln ⁡ ⁡ cos ⁡ x f π 3 = 1 2 - ln ⁡ 1 2 = 1 2 + ln ⁡ 2 ⇒ a = b = 2 ∴ a + b = 4