Calculer-0-dt-e-t-e-t-2-a-2-

Question Number 195203 by Erico last updated on 27/Jul/23

Calculer {\int_{0}}^{+ \infty} \frac{dt}{{(e^{t} - e^{- t})}^{2} + a^{2}}

Answered by Frix last updated on 27/Jul/23

\underset{0}{\int^{\infty}} \frac{d t}{{(e^{t} - e^{- t})}^{2} + a^{2}} \overset{[u = e^{2 t}]}{=}

= \frac{1}{2} \underset{1}{\int^{\infty}} \frac{d u}{u^{2} + (a^{2} - 2) u + 1} =

= \frac{1}{2 a \sqrt{a^{2} - 4}} {[\ln ∣ \frac{2 u + a^{2} - 2 + a \sqrt{a^{2} - 4}}{2 u + a^{2} - 2 - a \sqrt{a^{2} - 4}} ∣]}_{1}^{\infty} =

= \frac{1}{2 a \sqrt{a^{2} - 4}} \ln ∣ \frac{a - \sqrt{a^{2} - 4}}{a + \sqrt{a^{2} - 4}} ∣

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