Question

ajudm m irmaos, como se resolve " INTEGRAL DE dx/(e^x+1)"?

Answer 1

$\int \frac{dx}{(e^x+1)^2}$

substituição
$u = e^x+1\to (u-1)=e^x\\\\ \frac{du}{dx}=e^x \\\\ \frac{du}{e^x} =dx\\\\ \frac{du}{(u-1)}=dx$

ficando
$\int \frac{1}{u}* \frac{du}{(u-1)}$

decompondo em frações parciais
$\frac{1}{u(u-1)}= \frac{A}{u}+ \frac{B}{(u-1)} \\\\ \frac{1}{u(u-1)} = \frac{A(u-1)+Bu}{u(u-1)} \\\\1=A(u-1)+Bu\\\\\text{para u=0}\\\\1=A(0-1)+B*0\\A=-1\\\\\\\text{para u =1}\\\\1=A(1-1)+B*1\\\\1=B$

então
$\frac{1}{u*(u-1)} = \frac{-1}{u} + \frac{1}{(u-1)}$

ficando
$\int \frac{-1}{u} + \frac{1}{(u-1)} du\\\\= -ln(u)+ln(u-1) +C \\\\= -ln(e^x+1)+ln(e^x+1 -1)+C\\\\=-ln(e^x+1)+ln(e^x)+C\\\\= -ln(e^x+1)+x+C$