The value of \(\int_{0}^{\frac{\pi}{4}}\displaystyle\tan^{2}x\,dx\) is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(1-\displaystyle\frac{\pi}{4}\)
Using the identity \(\tan^{2}x = \sec^{2}x - 1\),
\( \displaystyle \int_{0}^{\pi/4}\tan^{2}x\,dx = \int_{0}^{\pi/4}(\sec^{2}x - 1)\,dx \)
\( \displaystyle = \left[\tan x - x\right]_{0}^{\pi/4} \)
\( \displaystyle = \left(1 - \frac{\pi}{4}\right) - 0 = 1 - \frac{\pi}{4} \)