Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| Definite integral | Value | ||
| (A) | \(\int_{0}^{\frac{\pi}{2}}\displaystyle\sin x\,dx\) | (I) | \(\displaystyle\frac{\pi}{3}\) |
| (B) | \(\int_{1}^{e^{2}}\displaystyle\frac{dx}{x}\) | (II) | \(\displaystyle\frac{5}{6}\) |
| (C) | \(\int_{0}^{\sqrt{3}}\displaystyle\frac{dx}{1+x^{2}}\) | (III) | 1 |
| (D) | \(\int_{0}^{1}\displaystyle x(1+x)\,dx\) | (IV) | 2 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A)-(III), (B)-(IV), (C)-(I), (D)-(II)
(A) \(\int_{0}^{\pi/2}\sin x\,dx\)
This matches (III) 1:
\( \displaystyle \left[-\cos x\right]_{0}^{\pi/2} = -0 + 1 = 1 \)
(B) \(\int_{1}^{e^{2}}\frac{dx}{x}\)
This matches (IV) 2:
\( \displaystyle \left[\log_e x\right]_{1}^{e^{2}} = 2 - 0 = 2 \)
(C) \(\int_{0}^{\sqrt{3}}\frac{dx}{1+x^{2}}\)
This matches (I) \(\frac{\pi}{3}\):
\( \displaystyle \left[\tan^{-1}x\right]_{0}^{\sqrt{3}} = \frac{\pi}{3} - 0 = \frac{\pi}{3} \)
(D) \(\int_{0}^{1}x(1+x)\,dx\)
This matches (II) \(\frac{5}{6}\):
\( \displaystyle \int_{0}^{1}(x+x^{2})\,dx = \frac{1}{2} + \frac{1}{3} = \frac{5}{6} \)