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NDA 2023 Maths · Mathematics · Integrals · medium

Using the substitution u=1+x^2, the indefinite integral of x√(1+x^2 with respect to x is

Single correct option

Using the substitution u=1+x2u=1+x^2, the indefinite integral of x1+x2x\sqrt{1+x^2} with respect to xx is
  1. A. 23(1+x2)3/2+C\dfrac23(1+x^2)^{3/2}+C
  2. B. 12(1+x2)3/2+C\dfrac12(1+x^2)^{3/2}+C
  3. C. 13(1+x2)1/2+C\dfrac13(1+x^2)^{1/2}+C
  4. D. 13(1+x2)3/2+C\dfrac13(1+x^2)^{3/2}+C
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Answer: (D) 13(1+x2)3/2+C\dfrac13(1+x^2)^{3/2}+C

Put u=1+x2u=1+x^2, du=2x dxdu=2x\,dx: the integral is 12∫u1/2du=12⋅23u3/2=13(1+x2)3/2\dfrac12\int u^{1/2}du=\dfrac12\cdot\dfrac23u^{3/2}=\dfrac13(1+x^2)^{3/2}. The coefficient 23\dfrac23 forgets the factor 12\dfrac12 coming from dudu.
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