Integral de $$$\frac{1}{\sqrt{x^{2} + 4}}$$$

Encontre $$$\int \frac{1}{\sqrt{x^{2} + 4}}\, dx$$$.

Seja $$$x=2 \sinh{\left(u \right)}$$$.

Então $$$dx=\left(2 \sinh{\left(u \right)}\right)^{\prime }du = 2 \cosh{\left(u \right)} du$$$ (os passos podem ser vistos »).

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

Logo,

$$$\frac{1}{\sqrt{x^{2} + 4}} = \frac{1}{\sqrt{4 \sinh^{2}{\left( u \right)} + 4}}$$$

Use a identidade $$$\sinh^{2}{\left( u \right)} + 1 = \cosh^{2}{\left( u \right)}$$$:

$$$\frac{1}{\sqrt{4 \sinh^{2}{\left( u \right)} + 4}}=\frac{1}{2 \sqrt{\sinh^{2}{\left( u \right)} + 1}}=\frac{1}{2 \sqrt{\cosh^{2}{\left( u \right)}}}$$$

$$$\frac{1}{2 \sqrt{\cosh^{2}{\left( u \right)}}} = \frac{1}{2 \cosh{\left( u \right)}}$$$

Assim,

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

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

$${\color{red}{\int{1 d u}}} = {\color{red}{u}}$$

Recorde que $$$u=\operatorname{asinh}{\left(\frac{x}{2} \right)}$$$:

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

Portanto,

$$\int{\frac{1}{\sqrt{x^{2} + 4}} d x} = \operatorname{asinh}{\left(\frac{x}{2} \right)}$$

Adicione a constante de integração:

$$\int{\frac{1}{\sqrt{x^{2} + 4}} d x} = \operatorname{asinh}{\left(\frac{x}{2} \right)}+C$$

$$$\int \frac{1}{\sqrt{x^{2} + 4}}\, dx = \operatorname{asinh}{\left(\frac{x}{2} \right)} + C$$$A