Question

Qual a resolução dessa integral

Answer 1

$\int\ {\frac{e^x}{1+e^x}} \, dx$

$u=e^x$

$du=e^xdx$

$dx=\frac{du}{e^x}$

$\int\ {\frac{e^x}{e^x(1+u)}} \, du$

$\int\ {\frac{1}{1+u}} \, du$

$v=1+u$

$dv=du$

$\int\ {\frac{1}{v}} \, dv$

$ln|v|$

$v=1+u$

$ln|1+u|$

$u=e^x$

$\boxed{ln|1+e^x|+C}$