Question

integral de e^x dividido por (1-e^x)^2 dx
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Answer 1

$\displaystyle \sf \int \frac{e^{x}}{(1-e^{x})^2}dx \\\\\\ \text{Fa{\c c}amos} : \\\\ u = 1-e^{x} \to du = -e^{x}dx\to -du=e^{x}dx \\\\\ Da{\'i}} :\\\\ \int \frac{-du}{u^2} \to -\int u^{-2}du \\\\\\ -\frac{u^{(-2+1)}}{\displaystyle \sf -2+1} +C \\\\\\ -\frac{u^{-1}}{-1} + C \\\\\\ \frac{1}{u} + C \to \frac{1}{1-e^x}+C \\\\ Portanto : \\\\ \boxed{\sf \ \int \frac{e^{x}}{(1-e^{x})^2}dx = \frac{1}{1-e^x}+C \ }\checkmark$