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