Integral de $$$\frac{x}{\sqrt{x} - 1}$$$

Encontre $$$\int \frac{x}{\sqrt{x} - 1}\, dx$$$.

Seja $$$u=\sqrt{x}$$$.

Então $$$du=\left(\sqrt{x}\right)^{\prime }dx = \frac{1}{2 \sqrt{x}} dx$$$ (veja os passos »), e obtemos $$$\frac{dx}{\sqrt{x}} = 2 du$$$.

Assim,

$${\color{red}{\int{\frac{x}{\sqrt{x} - 1} d x}}} = {\color{red}{\int{\frac{2 u^{3}}{u - 1} d u}}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=2$$$ e $$$f{\left(u \right)} = \frac{u^{3}}{u - 1}$$$:

$${\color{red}{\int{\frac{2 u^{3}}{u - 1} d u}}} = {\color{red}{\left(2 \int{\frac{u^{3}}{u - 1} d u}\right)}}$$

Como o grau do numerador não é menor que o grau do denominador, realize a divisão longa de polinômios (os passos podem ser vistos »):

$$2 {\color{red}{\int{\frac{u^{3}}{u - 1} d u}}} = 2 {\color{red}{\int{\left(u^{2} + u + 1 + \frac{1}{u - 1}\right)d u}}}$$

Integre termo a termo:

$$2 {\color{red}{\int{\left(u^{2} + u + 1 + \frac{1}{u - 1}\right)d u}}} = 2 {\color{red}{\left(\int{1 d u} + \int{u d u} + \int{u^{2} d u} + \int{\frac{1}{u - 1} d u}\right)}}$$

Aplique a regra da constante $$$\int c\, du = c u$$$ usando $$$c=1$$$:

$$2 \int{u d u} + 2 \int{u^{2} d u} + 2 \int{\frac{1}{u - 1} d u} + 2 {\color{red}{\int{1 d u}}} = 2 \int{u d u} + 2 \int{u^{2} d u} + 2 \int{\frac{1}{u - 1} d u} + 2 {\color{red}{u}}$$

Aplique a regra da potência $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=1$$$:

$$2 u + 2 \int{u^{2} d u} + 2 \int{\frac{1}{u - 1} d u} + 2 {\color{red}{\int{u d u}}}=2 u + 2 \int{u^{2} d u} + 2 \int{\frac{1}{u - 1} d u} + 2 {\color{red}{\frac{u^{1 + 1}}{1 + 1}}}=2 u + 2 \int{u^{2} d u} + 2 \int{\frac{1}{u - 1} d u} + 2 {\color{red}{\left(\frac{u^{2}}{2}\right)}}$$

Aplique a regra da potência $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ com $$$n=2$$$:

$$u^{2} + 2 u + 2 \int{\frac{1}{u - 1} d u} + 2 {\color{red}{\int{u^{2} d u}}}=u^{2} + 2 u + 2 \int{\frac{1}{u - 1} d u} + 2 {\color{red}{\frac{u^{1 + 2}}{1 + 2}}}=u^{2} + 2 u + 2 \int{\frac{1}{u - 1} d u} + 2 {\color{red}{\left(\frac{u^{3}}{3}\right)}}$$

Seja $$$v=u - 1$$$.

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

A integral pode ser reescrita como

$$\frac{2 u^{3}}{3} + u^{2} + 2 u + 2 {\color{red}{\int{\frac{1}{u - 1} d u}}} = \frac{2 u^{3}}{3} + u^{2} + 2 u + 2 {\color{red}{\int{\frac{1}{v} d v}}}$$

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

$$\frac{2 u^{3}}{3} + u^{2} + 2 u + 2 {\color{red}{\int{\frac{1}{v} d v}}} = \frac{2 u^{3}}{3} + u^{2} + 2 u + 2 {\color{red}{\ln{\left(\left|{v}\right| \right)}}}$$

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

$$\frac{2 u^{3}}{3} + u^{2} + 2 u + 2 \ln{\left(\left|{{\color{red}{v}}}\right| \right)} = \frac{2 u^{3}}{3} + u^{2} + 2 u + 2 \ln{\left(\left|{{\color{red}{\left(u - 1\right)}}}\right| \right)}$$

Recorde que $$$u=\sqrt{x}$$$:

$$2 \ln{\left(\left|{-1 + {\color{red}{u}}}\right| \right)} + 2 {\color{red}{u}} + {\color{red}{u}}^{2} + \frac{2 {\color{red}{u}}^{3}}{3} = 2 \ln{\left(\left|{-1 + {\color{red}{\sqrt{x}}}}\right| \right)} + 2 {\color{red}{\sqrt{x}}} + {\color{red}{\sqrt{x}}}^{2} + \frac{2 {\color{red}{\sqrt{x}}}^{3}}{3}$$

Portanto,

$$\int{\frac{x}{\sqrt{x} - 1} d x} = \frac{2 x^{\frac{3}{2}}}{3} + 2 \sqrt{x} + x + 2 \ln{\left(\left|{\sqrt{x} - 1}\right| \right)}$$

Adicione a constante de integração:

$$\int{\frac{x}{\sqrt{x} - 1} d x} = \frac{2 x^{\frac{3}{2}}}{3} + 2 \sqrt{x} + x + 2 \ln{\left(\left|{\sqrt{x} - 1}\right| \right)}+C$$

$$$\int \frac{x}{\sqrt{x} - 1}\, dx = \left(\frac{2 x^{\frac{3}{2}}}{3} + 2 \sqrt{x} + x + 2 \ln\left(\left|{\sqrt{x} - 1}\right|\right)\right) + C$$$A