Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| Definite integral | Value | ||
| (A) | \(\int_{0}^{1}\displaystyle xe^{x^{2}}\,dx\) | (I) | \(\displaystyle\frac{1}{3}\) |
| (B) | \(\int_{0}^{\frac{\pi}{2}}\displaystyle\sin x\cos^{2}x\,dx\) | (II) | \(\log_e 2\) |
| (C) | \(\int_{e}^{e^{2}}\displaystyle\frac{dx}{x\log_e x}\) | (III) | \(\displaystyle\frac{e-1}{2}\) |
| (D) | \(\int_{0}^{2}\displaystyle\frac{x}{1+x^{2}}\,dx\) | (IV) | \(\displaystyle\frac{1}{2}\log_e 5\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(III), (B)-(I), (C)-(II), (D)-(IV)
Each integral reduces to a simple one by a substitution with new limits.
(A) \(\int_{0}^{1}xe^{x^{2}}\,dx\)
This matches (III) \(\frac{e-1}{2}\). With \(t = x^{2}\), \(dt = 2x\,dx\), limits 0 to 1:
\( \displaystyle \frac{1}{2}\int_{0}^{1}e^{t}\,dt = \frac{e-1}{2} \)
(B) \(\int_{0}^{\pi/2}\sin x\cos^{2}x\,dx\)
This matches (I) \(\frac{1}{3}\). With \(t = \cos x\), \(dt = -\sin x\,dx\), limits 1 to 0:
\( \displaystyle \int_{0}^{1}t^{2}\,dt = \frac{1}{3} \)
(C) \(\int_{e}^{e^{2}}\frac{dx}{x\log_e x}\)
This matches (II) \(\log_e 2\). With \(t = \log_e x\), \(dt = \frac{dx}{x}\), limits 1 to 2:
\( \displaystyle \int_{1}^{2}\frac{dt}{t} = \log_e 2 - \log_e 1 = \log_e 2 \)
(D) \(\int_{0}^{2}\frac{x}{1+x^{2}}\,dx\)
This matches (IV) \(\frac{1}{2}\log_e 5\). With \(t = 1+x^{2}\), \(dt = 2x\,dx\), limits 1 to 5:
\( \displaystyle \frac{1}{2}\int_{1}^{5}\frac{dt}{t} = \frac{1}{2}\log_e 5 \)