Match List-I with List-II
| List-I | List-II | ||
| Integral | Value | ||
| (A) | \( \int_{0}^{3}[x]\,dx \), [x] is the greatest integer function. | (I) | 29 |
| (B) | \( \int_{0}^{1}x\,e^{x}\,dx \) | (II) | 9 |
| (C) | \( \int_{-5}^{5}|x-2|\,dx \) | (III) | 3 |
| (D) | \( \int_{0}^{3}x^{2}\,dx \) | (IV) | 1 |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) - (III), (B) - (IV), (C) - (I), (D) - (II)
(A) The greatest integer function [x] is 0, 1, 2 on the three unit intervals of [0, 3]:
\( \displaystyle \int_{0}^{3}[x]\,dx=0+1+2=3 \)
This matches (III).
(B) Integrating by parts:
\( \displaystyle \int_{0}^{1}x\,e^{x}\,dx=\left[xe^{x}-e^{x}\right]_{0}^{1} \)
\( \displaystyle =0-(-1)=1 \)
This matches (IV).
(C) Splitting at x = 2, where \( x-2 \) changes sign:
\( \displaystyle \int_{-5}^{5}|x-2|\,dx=\int_{-5}^{2}(2-x)\,dx+\int_{2}^{5}(x-2)\,dx \)
\( \displaystyle =\frac{49}{2}+\frac{9}{2} \)
\( \displaystyle =29 \)
This matches (I).
(D)
\( \displaystyle \int_{0}^{3}x^{2}\,dx=\frac{27}{3}=9 \)
This matches (II).
So the matching is (A) - (III), (B) - (IV), (C) - (I), (D) - (II).
Shortcut: For (C), the area under \( |x-2| \) is two triangles, with bases and heights 7 and 3:
\( \displaystyle \frac{1}{2}\cdot 7\cdot 7+\frac{1}{2}\cdot 3\cdot 3=29 \)