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27/03/2017

Um integral engraçado.

Já vi outras resoluções para isto. Vou apresentar a minha.
Problema: \[\int_{0}^{\pi} \sin x \ln {\cot \left( \frac{x}{2} \right)} dx \]
Possível resolução:
Vou começar por reescrever o integral na forma \[\int_{0}^{\pi} \sin x \ln \sqrt{ \frac{1+\cos (x)}{1-\cos(x)} } dx \] Depois faço a substituição $t=x+\frac{\pi}{2}$ obtendo \[\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \cos t \ln \sqrt{ \frac{1+\sin (t)}{1-\sin (t)} } dt.\] Como a função integranda é ímpar, então, no intervalo $\left]-\frac{\pi}{2};\frac{\pi}{2}\right[$ o integral vale zero.


PS:
Apresento abaixo o integral inicial, calculado numericamente numa CASIO CG20 (utilizando apenas a funções da calculadora, sem recorrer a programação...)

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