Integral de $$$\frac{1}{e^{x} + 1}$$$

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

Seja $$$u=e^{x}$$$.

Então $$$du=\left(e^{x}\right)^{\prime }dx = e^{x} dx$$$ (veja os passos »), e obtemos $$$e^{x} dx = du$$$.

A integral torna-se

$${\color{red}{\int{\frac{1}{e^{x} + 1} d x}}} = {\color{red}{\int{\frac{1}{u \left(u + 1\right)} d u}}}$$

Efetue a decomposição em frações parciais (os passos podem ser vistos »):

$${\color{red}{\int{\frac{1}{u \left(u + 1\right)} d u}}} = {\color{red}{\int{\left(- \frac{1}{u + 1} + \frac{1}{u}\right)d u}}}$$

Integre termo a termo:

$${\color{red}{\int{\left(- \frac{1}{u + 1} + \frac{1}{u}\right)d u}}} = {\color{red}{\left(\int{\frac{1}{u} d u} - \int{\frac{1}{u + 1} d u}\right)}}$$

A integral de $$$\frac{1}{u}$$$ é $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

$$- \int{\frac{1}{u + 1} d u} + {\color{red}{\int{\frac{1}{u} d u}}} = - \int{\frac{1}{u + 1} d u} + {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$

Seja $$$v=u + 1$$$.

Então $$$dv=\left(u + 1\right)^{\prime }du = 1 du$$$ (veja os passos »), e obtemos $$$du = dv$$$.

Logo,

$$\ln{\left(\left|{u}\right| \right)} - {\color{red}{\int{\frac{1}{u + 1} d u}}} = \ln{\left(\left|{u}\right| \right)} - {\color{red}{\int{\frac{1}{v} d v}}}$$

A integral de $$$\frac{1}{v}$$$ é $$$\int{\frac{1}{v} d v} = \ln{\left(\left|{v}\right| \right)}$$$:

$$\ln{\left(\left|{u}\right| \right)} - {\color{red}{\int{\frac{1}{v} d v}}} = \ln{\left(\left|{u}\right| \right)} - {\color{red}{\ln{\left(\left|{v}\right| \right)}}}$$

Recorde que $$$v=u + 1$$$:

$$\ln{\left(\left|{u}\right| \right)} - \ln{\left(\left|{{\color{red}{v}}}\right| \right)} = \ln{\left(\left|{u}\right| \right)} - \ln{\left(\left|{{\color{red}{\left(u + 1\right)}}}\right| \right)}$$

Recorde que $$$u=e^{x}$$$:

$$- \ln{\left(\left|{1 + {\color{red}{u}}}\right| \right)} + \ln{\left(\left|{{\color{red}{u}}}\right| \right)} = - \ln{\left(\left|{1 + {\color{red}{e^{x}}}}\right| \right)} + \ln{\left(\left|{{\color{red}{e^{x}}}}\right| \right)}$$

Portanto,

$$\int{\frac{1}{e^{x} + 1} d x} = x - \ln{\left(e^{x} + 1 \right)}$$

Adicione a constante de integração:

$$\int{\frac{1}{e^{x} + 1} d x} = x - \ln{\left(e^{x} + 1 \right)}+C$$

$$$\int \frac{1}{e^{x} + 1}\, dx = \left(x - \ln\left(e^{x} + 1\right)\right) + C$$$A