Question

calculo II integração
∫e×/1+e× dxI

Answer 1

$\displaystyle\int\dfrac{e^{x}}{1+e^{x}}dx$

Fazendo $u=1+e^{x}$, temos $du=e^{x}dx$, então

$\displaystyle\int\dfrac{e^{x}}{1+e^{x}}dx=\int\dfrac{1}{1+e^{x}}e^{x}dx\\\\\\\int\dfrac{e^{x}}{1+e^{x}}dx=\int\dfrac{1}{u}du\\\\\\\int\dfrac{e^{x}}{1+e^{x}}dx=ln|u|+C\\\\\\\int\dfrac{e^{x}}{1+e^{x}}dx=ln|1+e^{x}|+C$

Como $1+e^{x}~\textgreater~0~~\forall~x\in\mathbb{R}$:

$\boxed{\boxed{\int\dfrac{e^{x}}{1+e^{x}}dx=ln(1+e^{x})+C}}$