Question

The value of \(\displaystyle \int (\log\sec x)\tan x\,dx\) is

• A. \(\sec x+c\)
• B. \(\log\sec x+c\)
• C. \(\frac{1}{2}(\log\sec x)^2+c\)
• D. \(\log(\log\sec x)\)

Hint:

If an integral contains a function and its derivative, use substitution. Here, derivative of \(\log\sec x\) is \(\tan x\).

Answer 1

The Correct Option is C

We need to evaluate: \[ \int (\log\sec x)\tan x\,dx. \] We know that: \[ \frac{d}{dx}(\log\sec x)=\tan x. \] So let \[ t=\log\sec x. \] Then, \[ \frac{dt}{dx}=\tan x. \] Therefore, \[ dt=\tan x\,dx. \] Now substitute in the integral: \[ \int (\log\sec x)\tan x\,dx=\int t\,dt. \] Now integrate: \[ \int t\,dt=\frac{t^2}{2}+c. \] Substitute back \[ t=\log\sec x. \] So, \[ \frac{t^2}{2}+c = \frac{1}{2}(\log\sec x)^2+c. \] Hence, \[ \int (\log\sec x)\tan x\,dx = \frac{1}{2}(\log\sec x)^2+c. \]