Match List-I with List-II
| List-I | List-II | ||
| Definite Integral | Value | ||
| (A) | \( \int_{1}^{2}\displaystyle e^{x}\,dx \) | (I) | \( \displaystyle\frac{e^2(e^2-1)}{2} \) |
| (B) | \( \int_{0}^{1}\displaystyle e^{x}\,dx \) | (II) | \( \displaystyle\frac{(e^2-1)}{2} \) |
| (C) | \( \int_{1}^{2}\displaystyle e^{2x}\,dx \) | (III) | \( e-1 \) |
| (D) | \( \int_{0}^{1}\displaystyle e^{2x}\,dx \) | (IV) | \( e(e-1) \) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) - (IV), (B) - (III), (C) - (I), (D) - (II)
Here \( \int e^{x}\,dx=e^{x} \) and \( \int e^{2x}\,dx=\frac{e^{2x}}{2} \).
(A) \( \int_{1}^{2}e^{x}\,dx \)
This matches (IV) \( e(e-1) \).
\( \displaystyle \Big[e^{x}\Big]_{1}^{2}=e^2-e \)
\( \displaystyle =e(e-1) \)
(B) \( \int_{0}^{1}e^{x}\,dx \)
This matches (III) \( e-1 \).
\( \displaystyle \Big[e^{x}\Big]_{0}^{1}=e-1 \)
(C) \( \int_{1}^{2}e^{2x}\,dx \)
This matches (I) \( \frac{e^2(e^2-1)}{2} \).
\( \displaystyle \left[\frac{e^{2x}}{2}\right]_{1}^{2}=\frac{e^4-e^2}{2} \)
\( \displaystyle =\frac{e^2(e^2-1)}{2} \)
(D) \( \int_{0}^{1}e^{2x}\,dx \)
This matches (II) \( \frac{(e^2-1)}{2} \).
\( \displaystyle \left[\frac{e^{2x}}{2}\right]_{0}^{1}=\frac{e^2-1}{2} \)
So the matching is (A)-(IV), (B)-(III), (C)-(I), (D)-(II).