The value of \(\int_{0}^{\frac{\pi}{4}}\displaystyle e^{\tan x}\sec^{2}x\,dx\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(e - 1\)
Let \(t = \tan x\). Then the substitution gives
\(dt = \sec^{2}x\,dx\)
The limits change: at \(x = 0\), \(t = \tan 0 = 0\); at \(x = \frac{\pi}{4}\), \(t = \tan\frac{\pi}{4} = 1\).
\( \displaystyle \int_{0}^{\frac{\pi}{4}}e^{\tan x}\sec^{2}x\,dx = \int_{0}^{1}e^{t}\,dt \)
\( \displaystyle = \Big[e^{t}\Big]_{0}^{1} \)
\( = e^{1} - e^{0} \)
\( = e - 1 \)