JEE Main20198 Apr 2019Morning ShiftMathematicsIndefinite IntegrationActual
∫ s i n 5 x 2 s i n x 2 d x , is equal to
Options
- Ax + 2 sin x + sin 2 x + c
- B2 x + sin x + sin 2 x + c
- Cx + 2 sin x + 2 sin 2 x + c
- D  2 x + sin x + 2 sin 2 x + c
Correct answer
A. x + 2 sin x + sin 2 x + c
Step-by-step solution
We have, I = ∫ s i n 5 x 2 s i n x 2 d x = ∫ s i n 2 x + x 2 s i n x 2 d x Using sin A + B = sin A cos B + cos A sin B = ∫ s i n 2 x c o s x 2 + c o s 2 x s i n x 2 s i n x 2 d x = ∫ 2   sin x   cos x c o s x 2 s i n x 2 + c o s   2 x d x = ∫ 2 2 sin x 2 cos x 2 cos x   cos x 2 sin x 2 + cos   2 x d x = ∫ 4 cos 2 x 2 cos x + cos 2 x d x Using cos 2 x = 2 cos 2 x - 1 ⇒ 2 cos 2 x = 1 + cos 2 x = ∫ 2 1 + cos x cos x + cos 2 x d x = ∫ 2 cos