Question:
The value of
\[
\int (\log\sec x)\tan x\,dx
\]
is
The value of
\[
\int (\log\sec x)\tan x\,dx
\]
is
Show Hint
Use substitution when one factor is the derivative of another factor.
Updated On: May 7, 2026
- \(\sec x+c\)
- \(\log\sec x+c\)
- \(\frac{1}{2}(\log\sec x)^2+c\)
- \(\log(\log\sec x)\)
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The Correct Option is C
Solution and Explanation
Step 1: Let: \[ t=\log\sec x \]
Step 2: Differentiate: \[ dt=\tan x\,dx \]
Step 3: Therefore: \[ \int(\log\sec x)\tan x\,dx = \int t\,dt \]
Step 4: \[ \int t\,dt=\frac{t^2}{2}+c \]
Step 5: Substitute \(t=\log\sec x\): \[ \frac{1}{2}(\log\sec x)^2+c \] \[ \boxed{\frac{1}{2}(\log\sec x)^2+c} \]
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