JEE PYQ: Indefinite Integration Question 19
Question 19 - 2019 (08 Apr Shift 1)
$\int \frac{\sin\frac{5x}{2}}{\sin\frac{x}{2}},dx$ is equal to (where $c$ is a constant of integration):
(1) $2x + \sin x + 2\sin 2x + c$
(2) $x + 2\sin x + 2\sin 2x + c$
(3) $x + 2\sin x + \sin 2x + c$
(4) $2x + \sin x + \sin 2x + c$
Show Answer
Answer: (3)
Solution
$\frac{\sin\frac{5x}{2}}{\sin\frac{x}{2}} = \frac{2\cos\frac{x}{2}\sin\frac{5x}{2}}{2\cos\frac{x}{2}\sin\frac{x}{2}} = \frac{\sin 3x + \sin 2x}{\sin x} = \frac{3\sin x - 4\sin^3 x + 2\sin x\cos x}{\sin x} = 3 - 4\sin^2 x + 2\cos x = 1 + 2\cos 2x + 2\cos x$. Integrating: $x + \sin 2x + 2\sin x + c$.
BETA
