Question

\(\displaystyle \int_{\pi/6}^{\pi/4}\tan\theta\,d\theta=\ \ ?\)

Hint:

$\int\tan\theta\,d\theta=\ln|\sec\theta|$; definite values often stay as logs.

Answer 1

Use \(\displaystyle\int \tan\theta\,d\theta=\ln|\sec\theta|+C\). Evaluate: \[ \ln(\sec\tfrac{\pi}{4})-\ln(\sec\tfrac{\pi}{6}) =\ln(\sqrt2)-\ln\!\Big(\tfrac{2}{\sqrt3}\Big) =\ln\!\Big(\tfrac{\sqrt6}{2}\Big). \] This value is not \(0,1,2,\) or \(3\). Hence none of the given choices matches.