If the integral I = ∫ tan ⁡ x 5 + 7 tan 2 ⁡ x d x = k ln ⁡ f x + C (where C is the integration constant) and f 0 = 5 , then the value of f π 4 is equal to

Step-by-step solution

I = ∫ sin ⁡ x cos ⁡ x 5 cos 2 ⁡ x + 7 sin 2 ⁡ x d x Let sin 2 ⁡ x = t ⇒ sin ⁡ x cos ⁡ x d x = d t 2 I = 1 2 ∫ d t 5 1 - t + 7 t = 1 2 ∫ d t 5 + 2 t = 1 4 ln ⁡ 5 + 2 t + C = 1 4 ln ⁡ 5 + 2 sin 2 ⁡ x + C ∴ f x = 5 + 2 sin 2 ⁡ x ⇒ f π 4 = 6