Match List-I with List-II (C is a constant of integration)
| List-I | List-II | ||
| (A) | \(\int x\sin x\,dx\) | (I) | \(x\sin x+\cos x+C\) |
| (B) | \(\int x\cos x\,dx\) | (II) | \(\sin x-x\cos x+C\) |
| (C) | \(\int x\sec^2x\,dx\) | (III) | \(-x\cot x+\log|\sin x|+C\) |
| (D) | \(\int x\,\mathrm{cosec}^2x\,dx\) | (IV) | \(x\tan x+\log|\cos x|+C\) |
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A)-(II), (B)-(I), (C)-(IV), (D)-(III)
In each integral x is the first function in integration by parts, \(\int x\,v\,dx = x\int v\,dx-\int\left(\int v\,dx\right)dx\).
(A) \(\int x\sin x\,dx\)
This matches (II) \(\sin x-x\cos x+C\).
\( \int x\sin x\,dx = -x\cos x+\int\cos x\,dx = -x\cos x+\sin x+C \)
(B) \(\int x\cos x\,dx\)
This matches (I) \(x\sin x+\cos x+C\).
\( \int x\cos x\,dx = x\sin x-\int\sin x\,dx = x\sin x+\cos x+C \)
(C) \(\int x\sec^2x\,dx\)
This matches (IV) \(x\tan x+\log|\cos x|+C\).
\( \int x\sec^2x\,dx = x\tan x-\int\tan x\,dx \)
\( = x\tan x+\log|\cos x|+C \)
(D) \(\int x\,\mathrm{cosec}^2x\,dx\)
This matches (III) \(-x\cot x+\log|\sin x|+C\).
\( \int x\,\mathrm{cosec}^2x\,dx = -x\cot x+\int\cot x\,dx \)
\( = -x\cot x+\log|\sin x|+C \)