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NDA 2 2024 Maths, question 100

Topic: Integral calculus

Passage · Q99 to Q100

Let 2∫x2−1x2+1dx=U(x)V(x)−3ln⁡2\int \frac{x^{2}-1}{\sqrt{x^{2}+1}} dx=U(x)V(x)-3\ln {U(x)+V(x)U(x)+V(x)} +c+c

Q100

What is U(x)V(x)U(x)V(x) equal to?

  1. (a) x2+x4\sqrt{x^{2}+x^{4}}Correct answer
  2. (b) x+x3\sqrt{x+x^{3}}
  3. (c) x2+x42\frac{\sqrt{x^{2}+x^{4}}}{2}
  4. (d) 2x2+x42\sqrt{x^{2}+x^{4}}

Solution.

Answer: (a) x2+x4\sqrt{x^{2}+x^{4}}

  1. From question 99, U=xU = x and V=x2+1V = \sqrt{x^{2} + 1}.
  2. UV=xx2+1=x2(x2+1)=x2+x4UV = x\sqrt{x^{2} + 1} = \sqrt{x^{2}(x^{2} + 1)} = \sqrt{x^{2} + x^{4}}.

Trap: Moving xx inside the square root holds for x≥0x \geq 0; for negative xx the sign differs.