Match List-I with List-II
| List-I | List-II | ||
|---|---|---|---|
| Definite integral | Value | ||
| (A) | \(\int_{0}^{\frac{\pi}{2}}\displaystyle\sin^{2}x\,dx\) | (I) | \(\displaystyle\frac{4}{3}\) |
| (B) | \(\int_{0}^{\frac{\pi}{2}}\displaystyle\cos^{3}x\,dx\) | (II) | \(\displaystyle\frac{\pi}{4}\) |
| (C) | \(\int_{0}^{\frac{\pi}{4}}\displaystyle\sec^{4}x\,dx\) | (III) | \(\displaystyle\frac{1}{2}\) |
| (D) | \(\int_{0}^{\frac{\pi}{2}}\displaystyle\sin x\cos x\,dx\) | (IV) | \(\displaystyle\frac{2}{3}\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A)-(II), (B)-(IV), (C)-(I), (D)-(III)
(A) \(\int_{0}^{\pi/2}\sin^{2}x\,dx\)
This matches (II) \(\frac{\pi}{4}\). Using \(\sin^{2}x = \frac{1-\cos 2x}{2}\):
\( \displaystyle \frac{1}{2}\left[x - \frac{\sin 2x}{2}\right]_{0}^{\pi/2} = \frac{1}{2}\cdot\frac{\pi}{2} = \frac{\pi}{4} \)
(B) \(\int_{0}^{\pi/2}\cos^{3}x\,dx\)
This matches (IV) \(\frac{2}{3}\). Using \(\cos^{3}x = \cos x - \sin^{2}x\cos x\):
\( \displaystyle \left[\sin x - \frac{\sin^{3}x}{3}\right]_{0}^{\pi/2} = 1 - \frac{1}{3} = \frac{2}{3} \)
(C) \(\int_{0}^{\pi/4}\sec^{4}x\,dx\)
This matches (I) \(\frac{4}{3}\). Using \(\sec^{4}x = (1+\tan^{2}x)\sec^{2}x\) and \(t = \tan x\), limits 0 to 1:
\( \displaystyle \int_{0}^{1}(1+t^{2})\,dt = 1 + \frac{1}{3} = \frac{4}{3} \)
(D) \(\int_{0}^{\pi/2}\sin x\cos x\,dx\)
This matches (III) \(\frac{1}{2}\):
\( \displaystyle \left[\frac{\sin^{2}x}{2}\right]_{0}^{\pi/2} = \frac{1}{2} - 0 = \frac{1}{2} \)