Integral de $$$\frac{2}{x \sqrt{x^{2} - 4}}$$$

Encontre $$$\int \frac{2}{x \sqrt{x^{2} - 4}}\, dx$$$.

Aplique a regra do múltiplo constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ usando $$$c=2$$$ e $$$f{\left(x \right)} = \frac{1}{x \sqrt{x^{2} - 4}}$$$:

$${\color{red}{\int{\frac{2}{x \sqrt{x^{2} - 4}} d x}}} = {\color{red}{\left(2 \int{\frac{1}{x \sqrt{x^{2} - 4}} d x}\right)}}$$

Seja $$$u=\frac{1}{x}$$$.

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

Portanto,

$$2 {\color{red}{\int{\frac{1}{x \sqrt{x^{2} - 4}} d x}}} = 2 {\color{red}{\int{\left(- \frac{1}{\sqrt{1 - 4 u^{2}}}\right)d u}}}$$

Aplique a regra do múltiplo constante $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ usando $$$c=-1$$$ e $$$f{\left(u \right)} = \frac{1}{\sqrt{1 - 4 u^{2}}}$$$:

$$2 {\color{red}{\int{\left(- \frac{1}{\sqrt{1 - 4 u^{2}}}\right)d u}}} = 2 {\color{red}{\left(- \int{\frac{1}{\sqrt{1 - 4 u^{2}}} d u}\right)}}$$

Seja $$$u=\frac{\sin{\left(v \right)}}{2}$$$.

Então $$$du=\left(\frac{\sin{\left(v \right)}}{2}\right)^{\prime }dv = \frac{\cos{\left(v \right)}}{2} dv$$$ (os passos podem ser vistos »).

Além disso, segue-se que $$$v=\operatorname{asin}{\left(2 u \right)}$$$.

Portanto,

$$$\frac{1}{\sqrt{1 - 4 u ^{2}}} = \frac{1}{\sqrt{1 - \sin^{2}{\left( v \right)}}}$$$

Use a identidade $$$1 - \sin^{2}{\left( v \right)} = \cos^{2}{\left( v \right)}$$$:

$$$\frac{1}{\sqrt{1 - \sin^{2}{\left( v \right)}}}=\frac{1}{\sqrt{\cos^{2}{\left( v \right)}}}$$$

Supondo que $$$\cos{\left( v \right)} \ge 0$$$, obtemos o seguinte:

$$$\frac{1}{\sqrt{\cos^{2}{\left( v \right)}}} = \frac{1}{\cos{\left( v \right)}}$$$

Assim,

$$- 2 {\color{red}{\int{\frac{1}{\sqrt{1 - 4 u^{2}}} d u}}} = - 2 {\color{red}{\int{\frac{1}{2} d v}}}$$

Aplique a regra da constante $$$\int c\, dv = c v$$$ usando $$$c=\frac{1}{2}$$$:

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

Recorde que $$$v=\operatorname{asin}{\left(2 u \right)}$$$:

$$- {\color{red}{v}} = - {\color{red}{\operatorname{asin}{\left(2 u \right)}}}$$

Recorde que $$$u=\frac{1}{x}$$$:

$$- \operatorname{asin}{\left(2 {\color{red}{u}} \right)} = - \operatorname{asin}{\left(2 {\color{red}{\frac{1}{x}}} \right)}$$

Portanto,

$$\int{\frac{2}{x \sqrt{x^{2} - 4}} d x} = - \operatorname{asin}{\left(\frac{2}{x} \right)}$$

Adicione a constante de integração:

$$\int{\frac{2}{x \sqrt{x^{2} - 4}} d x} = - \operatorname{asin}{\left(\frac{2}{x} \right)}+C$$

$$$\int \frac{2}{x \sqrt{x^{2} - 4}}\, dx = - \operatorname{asin}{\left(\frac{2}{x} \right)} + C$$$A