The value of ∫ cos ⁡ x sin 2 ⁡ x sin ⁡ x + cos ⁡ x d x is equal to

Step-by-step solution

Let, I =   ∫ cos ⁡ x sin 2 ⁡ x sin ⁡ x + cos ⁡ x d x On dividing numerator and denominator by cos 3 ⁡ x , and substituting tan x = t ,   sec 2 x d x = d t , we get, I =   ∫ d t t 2 1 + t =   ∫ - 1 t + 1 t 2 + 1 1 + t d t (using partial fraction) ⇒ I = − ∫ 1 t d t + ∫ t − 2 d t + ∫ 1 1 + t d t = - log ⁡ t - 1 t + log ⁡ 1 + t + C =   - 1 tan ⁡ x + log ⁡ 1 + tan ⁡ x tan ⁡ x +