Find the value of the definite integral ${\int }_0^{\pi /4}\tan xdx$.

Step-by-step solution

1. Recall that $\int \tan xdx=-\ln |\cos x|+C$ because derivative of $\cos x$ is $-\sin x$.
2. Apply the antiderivative to the limits: $-\ln |\cos x|{|}_0^{\pi /4}=-\ln (\cos \frac{\pi }{4})+\ln (\cos 0)$.
3. Evaluate the cosine values: $\cos \frac{\pi }{4}=\frac{\sqrt{2}}{2}$ and $\cos 0=1$.
4. Substitute: $-\ln \left(\frac{\sqrt{2}}{2}\right)+\ln 1=-\ln \left(\frac{1}{\sqrt{2}}\right)=\ln \sqrt{2}=\frac{1}{2}\ln 2$.
5. Thus the integral equals $\frac{1}{2}\ln 2$.

Concept tested: definite integral of trigonometric function

Published 31 August 2026 · Last updated 16 September 2026 · How this solution was prepared and checked

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