AP EAMCET201921 Apr 2019Evening ShiftMathematicsIndefinite IntegrationActual
∫ x + sin x 1 + cos x d x =
Options
- Alog e ( 1 + cos x ) + c
- Bx sin 2 x 2 + c
- Ctan x 2 + c
- Dx tan x 2 + c
Correct answer
D. x tan x 2 + c
Step-by-step solution
Consider I = ∫ x + sin x 1 + cos x d x ⇒ I = ∫ x 1 + cos x d x + ∫ sin x 1 + cos x d x Consider I = I 1 + I 2 Now, I 1 = ∫ x 1 + cos x d x = ∫ x d x 2 cos 2 x 2 ∵ 1 + cos θ = 2 cos 2 θ 2 ⇒ I 1 = ∫ x 2 sec 2 x 2 d x ⇒ I 1 = 2 x 2 tan x 2 - 2 log e sec x 2 + c 1 ∵ ∫ u v d x = u ∫ v d x - ∫ d u d x · ∫ v   d x d x ⇒ I 1 = x tan x 2 - log e sec 2 x 2 + c 1 Now, I 2 = ∫ sin x 1 + cos x d x Let 1 + co