NDA 1 2024 Maths, question 96

Topic: Integral calculus

Passage · Q95 to Q96

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

Q96

What is the value of $\beta$ ?

(a) $-\frac{2}{3}$
(b) $-\frac{1}{3}$
(c) $\frac{1}{3}$ Correct answer
(d) $\frac{2}{3}$

Solution.

Answer: (c) $\frac{1}{3}$

Rationalising the denominator produces the sum of the two square roots divided by 2.
Each integrates with coefficient (1/2)(2/3) = 1/3.
Therefore the coefficient β of (x − 1)^(3/2) is positive 1/3.