NDA 2 2024 Maths, question 99

Topic: Integral calculus

Passage · Q99 to Q100

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

Q99

What is $|U^{2}(x)-V^{2}(x)|$ equal to?

Solution.

Answer: (b) $1$

Differentiating checks that $2\int \frac{x^{2} - 1}{\sqrt{x^{2} + 1}}\,dx = x\sqrt{x^{2} + 1} - 3\ln\left(x + \sqrt{x^{2} + 1}\right) + c$ .
Matching the given form: $U = x$ and $V = \sqrt{x^{2} + 1}$ .
$|U^{2} - V^{2}| = |x^{2} - (x^{2} + 1)| = 1$ .

Trap: One equation cannot fix two functions in general; the matching $U = x$ , $V = \sqrt{x^{2} + 1}$ is the natural reading of the given form.