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NDA 2021 Maths · Mathematics · Integrals · medium

The antiderivative of (x+1)/(x^2+2x+3) with respect to x is

Single correct option

The antiderivative of x+1x2+2x+3\dfrac{x+1}{x^2+2x+3} with respect to xx is
  1. A. 12ln⁡(x2+2x+3)+C\dfrac12\ln(x^2+2x+3)+C
  2. B. ln⁡(x2+2x+3)+C\ln(x^2+2x+3)+C
  3. C. 12tan⁡−1(x+1)+C\dfrac12\tan^{-1}(x+1)+C
  4. D. 2ln⁡(x2+2x+3)+C2\ln(x^2+2x+3)+C
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Answer: (A) 12ln⁡(x2+2x+3)+C\dfrac12\ln(x^2+2x+3)+C

The numerator is half the derivative of the denominator (12⋅(2x+2)\tfrac12\cdot(2x+2)), so the integral is 12ln⁡(x2+2x+3)+C\tfrac12\ln(x^2+2x+3)+C. The denominator is always positive so no modulus is needed. The factor 12\tfrac12 is the common omission.
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Original question written in the NDA 2021 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