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NDA 2024 Maths · Mathematics · Integrals · hard

∫ e^x(sin x+cos x) dx is equal to (where C is a constant of integration)

Single correct option

∫ex(sin⁡x+cos⁡x) dx\displaystyle\int e^x(\sin x+\cos x)\,dx is equal to (where CC is a constant of integration)
  1. A. excos⁡x+Ce^x\cos x+C
  2. B. ex(sin⁡x−cos⁡x)+Ce^x(\sin x-\cos x)+C
  3. C. 2exsin⁡x+C2e^x\sin x+C
  4. D. exsin⁡x+Ce^x\sin x+C
Show answer and solution

Answer: (D) exsin⁡x+Ce^x\sin x+C

The integrand has the form ex(f(x)+f′(x))e^x(f(x)+f'(x)) with f(x)=sin⁡xf(x)=\sin x, so the integral is exf(x)+C=exsin⁡x+Ce^xf(x)+C=e^x\sin x+C. Check by differentiating: exsin⁡x+excos⁡xe^x\sin x+e^x\cos x. Trap: choosing f=cos⁡xf=\cos x would need f′=−sin⁡xf'=-\sin x, which gives the wrong sign.
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Original question written in the NDA 2024 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake