\(\int x\sec x\tan x\,dx\) is equal to (C is a constant of integration)
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(x\sec x-\log\left|\sec x+\tan x\right|+C\)
Using integration by parts with x as the first function and \(\sec x\tan x\) as the second:
\( \int\sec x\tan x\,dx = \sec x \)
\( \int x\sec x\tan x\,dx = x\sec x-\int\sec x\,dx \)
Using the standard result \(\int\sec x\,dx = \log\left|\sec x+\tan x\right|\):
\( \int x\sec x\tan x\,dx = x\sec x-\log\left|\sec x+\tan x\right|+C \)
Why the other options are incorrect
Option (1) \(x\sec x+\log\left|\sec x+\tan x\right|+C\) — Incorrect. The second integral of by parts is subtracted, not added.
Option (3) \(x\sec x-\log\left|\sec x\right|+C\) — Incorrect. \(\log\left|\sec x\right|\) is the integral of \(\tan x\), not of \(\sec x\).
Option (4) \(x\tan x-\log\left|\sec x+\tan x\right|+C\) — Incorrect. The integral of \(\sec x\tan x\) is \(\sec x\), so the first term is \(x\sec x\).