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JEE Main 2016 · Mathematics · Indefinite and Definite Integrals · hard

∫(cos 2x)/((sin x+cos x)^2 dx equals (where C is the constant of integration)

Single correct option

∫cos⁡2x(sin⁡x+cos⁡x)2 dx\displaystyle\int\frac{\cos2x}{(\sin x+\cos x)^2}\,dx equals (where CC is the constant of integration)
  1. A. −ln⁡∣sin⁡x+cos⁡x∣+C-\ln|\sin x+\cos x|+C
  2. B. ln⁡∣sin⁡x−cos⁡x∣+C\ln|\sin x-\cos x|+C
  3. C. ln⁡∣sin⁡x+cos⁡x∣+C\ln|\sin x+\cos x|+C
  4. D. 1sin⁡x+cos⁡x+C\dfrac{1}{\sin x+\cos x}+C
Show answer and solution

Answer: (C) ln⁡∣sin⁡x+cos⁡x∣+C\ln|\sin x+\cos x|+C

Use cos⁡2x=cos⁡2x−sin⁡2x=(cos⁡x−sin⁡x)(cos⁡x+sin⁡x)\cos2x=\cos^2x-\sin^2x=(\cos x-\sin x)(\cos x+\sin x). Cancelling one factor of (sin⁡x+cos⁡x)(\sin x+\cos x) gives ∫cos⁡x−sin⁡xcos⁡x+sin⁡x dx\displaystyle\int\frac{\cos x-\sin x}{\cos x+\sin x}\,dx. The numerator is exactly the derivative of the denominator, ddx(sin⁡x+cos⁡x)=cos⁡x−sin⁡x\frac{d}{dx}(\sin x+\cos x)=\cos x-\sin x, so the integral has the form ∫f′/f dx=ln⁡∣f∣\int f'/f\,dx=\ln|f|. The answer is ln⁡∣sin⁡x+cos⁡x∣+C\ln|\sin x+\cos x|+C. Differentiating this reproduces the integrand, which confirms the result.
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