Match List-I with List-II
| List-I | List-II | ||
| Definite Integral | Value | ||
| (A) | \( \int_{4}^{9}\displaystyle\frac{1}{\sqrt{x}}\,dx \) | (I) | \( \pi/4 \) |
| (B) | \( \int_{0}^{1/2}\displaystyle\frac{1}{\sqrt{1-x^2}}\,dx \) | (II) | \( 38/3 \) |
| (C) | \( \int_{0}^{1}\displaystyle\frac{1}{1+x^2}\,dx \) | (III) | 2 |
| (D) | \( \int_{4}^{9}\sqrt{x}\,dx \) | (IV) | \( \pi/6 \) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) - (III), (B) - (IV), (C) - (I), (D) - (II)
(A) \( \int_{4}^{9}\frac{1}{\sqrt{x}}\,dx \)
This matches (III) 2.
\( \displaystyle \int_{4}^{9}x^{-1/2}\,dx=\Big[2\sqrt{x}\Big]_{4}^{9} \)
\( \displaystyle =2(3-2)=2 \)
(B) \( \int_{0}^{1/2}\frac{1}{\sqrt{1-x^2}}\,dx \)
This matches (IV) π/6, by the standard integral \( \int\frac{dx}{\sqrt{1-x^2}}=\sin^{-1}x \).
\( \displaystyle \Big[\sin^{-1}x\Big]_{0}^{1/2}=\sin^{-1}\frac{1}{2}-0=\frac{\pi}{6} \)
(C) \( \int_{0}^{1}\frac{1}{1+x^2}\,dx \)
This matches (I) π/4, by the standard integral \( \int\frac{dx}{1+x^2}=\tan^{-1}x \).
\( \displaystyle \Big[\tan^{-1}x\Big]_{0}^{1}=\frac{\pi}{4}-0=\frac{\pi}{4} \)
(D) \( \int_{4}^{9}\sqrt{x}\,dx \)
This matches (II) 38/3.
\( \displaystyle \left[\frac{2}{3}x^{3/2}\right]_{4}^{9}=\frac{2}{3}(27-8) \)
\( \displaystyle =\frac{38}{3} \)
So the matching is (A)-(III), (B)-(IV), (C)-(I), (D)-(II).