JEE Main · 2019 · 08 Apr 2019 · Shift 1 · Indefinite Integrals

\int \frac{s i n \frac{5 x}{2}}{s i n \frac{x}{2}} d x

is equal to:
(where c is a constant of integration).

Answer: (C)

Solution

Multiply top and bottom by

cos \frac{x}{2}

so the denominator becomes

sin x

, using

2 sin \frac{5 x}{2} cos \frac{x}{2} = sin 3 x + sin 2 x

; the integrand turns into

\frac{sin 3 x + sin 2 x}{sin x}

. Writing

sin 3 x = 3 sin x - 4 sin^{3} x

and

sin 2 x = 2 sin x cos x

and dividing by

sin x

leaves

3 - 4 sin^{2} x + 2 cos x

. Replacing

sin^{2} x = \frac{1 - cos 2 x}{2}

gives

1 + 2 cos 2 x + 2 cos x

, which integrates to

x + sin 2 x + 2 sin x + c

JEE Main · 2019 · 08 Apr 2019 · Shift 1 · Single correct option

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