Match List-I with List-II
| List-I | List-II | ||
| (A) | \(\int_{0}^{2}\displaystyle 3x^{2}\,dx\) | (I) | 1 |
| (B) | \(\int_{1}^{e}\displaystyle\frac{2}{x}\,dx\) | (II) | 3 |
| (C) | \(\int_{0}^{\frac{\pi}{4}}\displaystyle\sec^{2}x\,dx\) | (III) | 8 |
| (D) | \(\int_{0}^{\log_e 4}\displaystyle e^{x}\,dx\) | (IV) | 2 |
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)-(I), (D)-(II)
Each integral is found from a standard antiderivative by the fundamental theorem of calculus.
(A) \(\int_{0}^{2}3x^{2}\,dx\)
This matches (III) 8.
\( \displaystyle \Big[x^{3}\Big]_{0}^{2} = 8 - 0 = 8 \)
(B) \(\int_{1}^{e}\frac{2}{x}\,dx\)
This matches (IV) 2.
\( \displaystyle \Big[2\log_e x\Big]_{1}^{e} = 2(1 - 0) = 2 \)
(C) \(\int_{0}^{\frac{\pi}{4}}\sec^{2}x\,dx\)
This matches (I) 1.
\( \displaystyle \Big[\tan x\Big]_{0}^{\frac{\pi}{4}} = 1 - 0 = 1 \)
(D) \(\int_{0}^{\log_e 4}e^{x}\,dx\)
This matches (II) 3.
\( \displaystyle \Big[e^{x}\Big]_{0}^{\log_e 4} = 4 - 1 = 3 \)