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

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

Seja $$$u=\sqrt{x - 1}$$$.

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

Portanto,

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

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

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

Reescreva e separe a fração:

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

Integre termo a termo:

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

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

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

A integral de $$$\frac{1}{u^{2} + 1}$$$ é $$$\int{\frac{1}{u^{2} + 1} d u} = \operatorname{atan}{\left(u \right)}$$$:

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

Recorde que $$$u=\sqrt{x - 1}$$$:

$$- 2 \operatorname{atan}{\left({\color{red}{u}} \right)} + 2 {\color{red}{u}} = - 2 \operatorname{atan}{\left({\color{red}{\sqrt{x - 1}}} \right)} + 2 {\color{red}{\sqrt{x - 1}}}$$

Portanto,

$$\int{\frac{\sqrt{x - 1}}{x} d x} = 2 \sqrt{x - 1} - 2 \operatorname{atan}{\left(\sqrt{x - 1} \right)}$$

Adicione a constante de integração:

$$\int{\frac{\sqrt{x - 1}}{x} d x} = 2 \sqrt{x - 1} - 2 \operatorname{atan}{\left(\sqrt{x - 1} \right)}+C$$

$$$\int \frac{\sqrt{x - 1}}{x}\, dx = \left(2 \sqrt{x - 1} - 2 \operatorname{atan}{\left(\sqrt{x - 1} \right)}\right) + C$$$A