Indefinite IntegrationhardFree

∫(sin2x+sin4x−sin6x)/(1+cos2x+cos4x+cos6x) | JEE

JEE Maths question with a full step-by-step solution.

Question
∫sin⁡2x+sin⁡4x−sin⁡6x1+cos⁡2x+cos⁡4x+cos⁡6x dx\displaystyle\int \dfrac{\sin 2x+\sin 4x-\sin 6x}{1+\cos 2x+\cos 4x+\cos 6x}\,dx equals
A13ln⁡∣sec⁡3x∣+12ln⁡∣sec⁡2x∣+ln⁡∣sec⁡x∣+C\dfrac{1}{3}\ln|\sec 3x|+\dfrac{1}{2}\ln|\sec 2x|+\ln|\sec x|+C
B13ln⁡∣sec⁡3x∣−12ln⁡∣sec⁡2x∣−ln⁡∣sec⁡x∣+C\dfrac{1}{3}\ln|\sec 3x|-\dfrac{1}{2}\ln|\sec 2x|-\ln|\sec x|+Ccorrect
CC−13ln⁡∣sec⁡3x∣+12ln⁡∣sec⁡2x∣+ln⁡∣sec⁡x∣C-\dfrac{1}{3}\ln|\sec 3x|+\dfrac{1}{2}\ln|\sec 2x|+\ln|\sec x|
DC−13ln⁡∣sec⁡3x∣−12ln⁡∣sec⁡2x∣+ln⁡∣sec⁡x∣C-\dfrac{1}{3}\ln|\sec 3x|-\dfrac{1}{2}\ln|\sec 2x|+\ln|\sec x|
Solution
Step 1: Factor the numerator. Pair sin⁡2x−sin⁡6x=−2cos⁡4xsin⁡2x\sin 2x-\sin 6x=-2\cos 4x\sin 2x and keep sin⁡4x=2sin⁡2xcos⁡2x\sin 4x=2\sin 2x\cos 2x:
Num=2sin⁡2x(cos⁡2x−cos⁡4x)=2sin⁡2x (2sin⁡3xsin⁡x)=4sin⁡xsin⁡2xsin⁡3x\text{Num}=2\sin 2x(\cos 2x-\cos 4x)=2\sin 2x\,(2\sin 3x\sin x)=4\sin x\sin 2x\sin 3x
 Step 2: Factor the denominator. Pair 1+cos⁡6x=2cos⁡23x1+\cos 6x=2\cos^{2}3x and cos⁡2x+cos⁡4x=2cos⁡3xcos⁡x\cos 2x+\cos 4x=2\cos 3x\cos x:
Den=2cos⁡23x+2cos⁡3xcos⁡x=2cos⁡3x(cos⁡3x+cos⁡x)\text{Den}=2\cos^{2}3x+2\cos 3x\cos x=2\cos 3x(\cos 3x+\cos x)
=2cos⁡3x (2cos⁡2xcos⁡x)=4cos⁡xcos⁡2xcos⁡3x=2\cos 3x\,(2\cos 2x\cos x)=4\cos x\cos 2x\cos 3x
 Step 3: Form the ratio:
NumDen=4sin⁡xsin⁡2xsin⁡3x4cos⁡xcos⁡2xcos⁡3x=tan⁡xtan⁡2xtan⁡3x\dfrac{\text{Num}}{\text{Den}}=\dfrac{4\sin x\sin 2x\sin 3x}{4\cos x\cos 2x\cos 3x}=\tan x\tan 2x\tan 3x
 Step 4: Use 3x=2x+x3x=2x+x, so tan⁡3x(1−tan⁡xtan⁡2x)=tan⁡x+tan⁡2x\tan 3x(1-\tan x\tan 2x)=\tan x+\tan 2x, giving
tan⁡xtan⁡2xtan⁡3x=tan⁡3x−tan⁡2x−tan⁡x\tan x\tan 2x\tan 3x=\tan 3x-\tan 2x-\tan x
 Step 5: Integrate each term with ∫tan⁡(ax) dx=1aln⁡∣sec⁡ax∣\displaystyle\int\tan(ax)\,dx=\dfrac{1}{a}\ln|\sec ax|:
I=∫(tan⁡3x−tan⁡2x−tan⁡x) dx=13ln⁡∣sec⁡3x∣−12ln⁡∣sec⁡2x∣−ln⁡∣sec⁡x∣+CI=\int(\tan 3x-\tan 2x-\tan x)\,dx=\dfrac{1}{3}\ln|\sec 3x|-\dfrac{1}{2}\ln|\sec 2x|-\ln|\sec x|+C
 Correct answer: (2)
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