Discussion

The value of \(\int_{0}^{\pi/3}\frac{\tan t}{\cos t}dt\) is equal to

Answer: E

The integral is 1.

\(\frac{\tan t}{\cos t} = \sec t\tan t\), whose antiderivative is \(\sec t\):

\[ \begin{aligned} \int_{0}^{\pi/3}\sec t\tan t\,dt \\ &= \sec\frac{\pi}{3} - \sec 0 \\ &= 2 - 1 \\ &= 1 \end{aligned} \]

2 forgets the lower limit, and \(\pm\frac{1}{2}\) or \(-2\) use \(\cos\) in place of \(\sec\).

The integrand is the derivative of \(\sec t\), which is the step to recognise: \(\frac{d}{dt}\sec t = \sec t\tan t\).

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