Question

A técnica de integração com logaritmo é basicamente:
$\displaystyle \int\,\frac{f'(x)}{f(x)}\,dx=\ln|f(x)|+C$
por exemplo:
$\displaystyle \int\,\frac{6x^2+2\cos x}{x^3+\sin x}\,dx=2\int\,\frac{3x^2+\cos x}{x^3+\sin x}\,dx\\\\ \frac{d}{dx}x^3+\sin x=3x^2+\cos x\\\\ 2\int\,\frac{3x^2+\cos x}{x^3+\sin x}\,dx=2\ln|x^3+\sin x|+C$

Integre: $\displaystyle \int\,\frac{\exp x}{\exp x-\exp (-x)}\,dx$
com esse método.

Answer 1

$I=\displaystyle\int\dfrac{\exp x}{\exp x-\exp(-x)}dx\\\\ I=\displaystyle\int\dfrac{\exp x}{\exp x-\exp(-x)}\cdot\dfrac{2\exp x}{2\exp x}dx\\\\ I=\dfrac{1}{2}\displaystyle\int\dfrac{2\exp(2x)}{\exp(2x)-1}dx$

Veja que $\dfrac{d}{dx}(\exp(2x)-1)=2\exp(2x)$. Então agora podemos aplicar o método:

$I=\dfrac{1}{2}\displaystyle\int\dfrac{2\exp(2x)}{\exp(2x)-1}dx\\\\\boxed{I=\dfrac{1}{2}\ln|\exp(2x)-1|+C}$