Match List-I with List-II
| List-I | List-II | ||
| (Functions) | (Integrals) where C is an arbitrary constant | ||
| (A) | \( \int\sqrt{4-x^{2}}\,dx \) | (I) | \( \displaystyle\frac{1}{2}\tan^{-1}\left(\displaystyle\frac{x}{2}\right)+C \) |
| (B) | \( \int\displaystyle\frac{1}{\sqrt{4-x^{2}}}\,dx \) | (II) | \( \displaystyle\frac{1}{4}\log\left|\displaystyle\frac{2+x}{2-x}\right|+C \) |
| (C) | \( \int\displaystyle\frac{1}{4-x^{2}}\,dx \) | (III) | \( \displaystyle\frac{x}{2}\sqrt{4-x^{2}}+2\sin^{-1}\left(\displaystyle\frac{x}{2}\right)+C \) |
| (D) | \( \int\displaystyle\frac{1}{4+x^{2}}\,dx \) | (IV) | \( \sin^{-1}\left(\displaystyle\frac{x}{2}\right)+C \) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A) - (III), (B) - (IV), (C) - (II), (D) - (I)
Each integral is a standard form, here with \( a=2 \).
(A) \( \int\sqrt{4-x^{2}}\,dx \)
This matches (III).
\( \displaystyle \int\sqrt{a^{2}-x^{2}}\,dx=\frac{x}{2}\sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2}\sin^{-1}\frac{x}{a}+C \)
(B) \( \int\frac{1}{\sqrt{4-x^{2}}}\,dx \)
This matches (IV).
\( \displaystyle \int\frac{dx}{\sqrt{a^{2}-x^{2}}}=\sin^{-1}\frac{x}{a}+C \)
(C) \( \int\frac{1}{4-x^{2}}\,dx \)
This matches (II).
\( \displaystyle \int\frac{dx}{a^{2}-x^{2}}=\frac{1}{2a}\log\left|\frac{a+x}{a-x}\right|+C \)
(D) \( \int\frac{1}{4+x^{2}}\,dx \)
This matches (I).
\( \displaystyle \int\frac{dx}{a^{2}+x^{2}}=\frac{1}{a}\tan^{-1}\frac{x}{a}+C \)
So the matching is (A)-(III), (B)-(IV), (C)-(II), (D)-(I).