Match List-I with List-II (Given that c is an arbitrary constant)
| List-I | List-II | ||
|---|---|---|---|
| (A) | \(\int\displaystyle\frac{dx}{\sqrt{a^2-x^2}}=\) | (I) | \(\log_e|x + \sqrt{x^2 - a^2}| + c\) |
| (B) | \(\int\displaystyle\sqrt{a^2-x^2}\, dx=\) | (II) | \(\displaystyle\sin^{-1}\frac{x}{a} + c\) |
| (C) | \(\int\displaystyle\sqrt{x^2 - a^2}\, dx =\) | (III) | \(\displaystyle\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}\sin^{-1}\frac{x}{a}+c\) |
| (D) | \(\int\displaystyle\frac{dx}{\sqrt{x^2-a^2}}=\) | (IV) | \(\displaystyle\frac{x}{a}\sqrt{x^2-a^2}-\frac{a^2}{2}\log_e|x + \sqrt{x^2 - a^2}| + c\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A)-(II), (B)-(III), (C)-(IV), (D)-(I)
Each item is a standard integral.
(A) \(\int\displaystyle\frac{dx}{\sqrt{a^2-x^2}}=\)
This matches (II):
\(\displaystyle \int\frac{dx}{\sqrt{a^2-x^2}} = \sin^{-1}\frac{x}{a} + c\)
(B) \(\int\displaystyle\sqrt{a^2-x^2}\, dx=\)
This matches (III):
\(\displaystyle \int\sqrt{a^2-x^2}\,dx = \frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\sin^{-1}\frac{x}{a} + c\)
(C) \(\int\displaystyle\sqrt{x^2 - a^2}\, dx =\)
This matches (IV), by elimination. The standard result is
\(\displaystyle \int\sqrt{x^2-a^2}\,dx = \frac{x}{2}\sqrt{x^2-a^2} - \frac{a^2}{2}\log|x + \sqrt{x^2-a^2}| + c\)
The printed (IV) has \(\frac{x}{a}\) in place of \(\frac{x}{2}\) in the first term, so as printed it equals this integral only when a = 2.
(D) \(\int\displaystyle\frac{dx}{\sqrt{x^2-a^2}}=\)
This matches (I):
\(\displaystyle \int\frac{dx}{\sqrt{x^2-a^2}} = \log|x + \sqrt{x^2-a^2}| + c\)
So the matching is (A)-(II), (B)-(III), (C)-(IV), (D)-(I).
Tip: in the integrals of \(\sqrt{a^2-x^2}\), \(\sqrt{x^2-a^2}\) and \(\sqrt{x^2+a^2}\), the first term is always \(\frac{x}{2}\) times the root, and the second term carries \(\frac{a^2}{2}\).
Note: It seems that there is a mistake in the original NTA paper, which prints (IV) as \(\frac{x}{a}\sqrt{x^2-a^2}-\frac{a^2}{2}\log_e|x + \sqrt{x^2 - a^2}| + c\) when it should read \(\frac{x}{2}\sqrt{x^2-a^2}-\frac{a^2}{2}\log_e|x + \sqrt{x^2 - a^2}| + c\). The question can still be solved because (A), (B) and (D) match (II), (III) and (I) exactly, and Option (4) is the only option containing these three matches.