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JEE Main 2019 · 8 April · Shift I

JEE Main 2019 Mathematics question on Indefinite Integration

∫sin⁡5x2sin⁡x2dx\int \dfrac{\sin \dfrac{5x}{2}}{\sin \dfrac{x}{2}} dx is equal to

(where, CC is a constant of integration)
  1. A.
    2x+sin⁡x+2sin⁡2x+C2x + \sin x + 2\sin 2x + C
  2. B.
    x+2sin⁡x+2sin⁡2x+Cx + 2\sin x + 2\sin 2x + C
  3. C.
    x+2sin⁡x+sin⁡2x+Cx + 2\sin x + \sin 2x + C
  4. D.
    2x+sin⁡x+sin⁡2x+C2x + \sin x + \sin 2x + C
Show answer and solution

Correct option: C

Solution

Use the identity: sin(5x/2)/sin(x/2) = 1 + 2cos(x) + 2cos(2x), which comes from the standard result that sin((2n+1)θ/2)/sin(θ/2) = 1 + 2cos θ + 2cos 2θ + ... + 2cos nθ (here n=2, θ=x).

So the integral becomes ∫(1 + 2cos x + 2cos 2x) dx = x + 2sin x + sin 2x + C.

[JEE Main 2019 · 8 April · Shift I]

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