Evaluate ∫₀^π/2 log(tan x) dx.

Definite Integrals & Area · Question 52 of 71

Question 52

HardOriginal question

Evaluate $\int_0^{\frac{\pi}{2}} \log(\tan x)\,dx$ .

- \frac{\pi \log 2}{4}

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0

Correct answerYour answer

\frac{\pi}{4}

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\frac{\pi}{2}

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CorrectNot quiteAnswer B

Correct answer: B

With $x \to \frac{\pi}{2} - x$ , $I = \int_0^{\frac{\pi}{2}} \log(\cot x)\,dx$ .
Adding: $2I = \int_0^{\frac{\pi}{2}} \log(\tan x\cot x)\,dx = \int_0^{\frac{\pi}{2}} \log 1\,dx = 0$ , so $I = 0$ .

Worth noting

$\frac{\pi}{4}$ is the answer for $\int_0^{\pi/2} \frac{\sin x}{\sin x + \cos x}\,dx$ , a different use of the same property.

Ten questions from Definite Integrals & Area, at your own pace.

Ten questions from Definite Integrals & Area, at your own pace.