Question

(15 PONTOS) Encontre a integral indefinida:
        
$\displaystyle\int{\dfrac{dx}{e^{x}+1}}$

Answer 1

A resposta segue anexa.

 

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23/02/2016 
Sepauto - SSRC
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Answer 2

$\Large\boxed{\begin{array}{l}\displaystyle\sf\int\dfrac{dx}{e^x+1}=\int\dfrac{e^x}{e^x(e^x+1)}dx\\\underline{\rm fac_{\!\!,}a}\\\sf u=e^x\implies du=e^x~dx\\\displaystyle\sf\int\dfrac{e^x}{e^x(e^x+1)}dx=\int\dfrac{du}{u(u+1)}\\\sf\dfrac{1}{u\cdot(u+1)}=\dfrac{A}{u}+\dfrac{B}{u+1}\\\\\sf A=\dfrac{1}{u+1}\bigg|_{u=0}=\dfrac{1}{0+1}=1\\\\\sf B=\dfrac{1}{u}\bigg|_{u=-1}=\dfrac{1}{-1}=-1\\\displaystyle\sf\int\dfrac{du}{u\cdot(u+1)}=\int\bigg(\dfrac{1}{u}-\dfrac{1}{u+1}\bigg)du\\\sf=\ell n|u|-\ell n|u+1|+k\\\\\displaystyle\sf\int\dfrac{du}{u\cdot(u+1)}du\\\sf=\ell n\bigg|\dfrac{u}{u+1}\bigg|+k\\\\\displaystyle\sf\int\dfrac{dx}{e^x+1}=\ell n\bigg|\dfrac{e^x}{e^x+1}\bigg|+k\end{array}}$