Integral de $$$\frac{\sqrt{\ln\left(x\right)}}{x}$$$

Encontre $$$\int \frac{\sqrt{\ln\left(x\right)}}{x}\, dx$$$.

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

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

Assim,

$${\color{red}{\int{\frac{\sqrt{\ln{\left(x \right)}}}{x} d x}}} = {\color{red}{\int{\sqrt{u} d u}}}$$

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

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

Recorde que $$$u=\ln{\left(x \right)}$$$:

$$\frac{2 {\color{red}{u}}^{\frac{3}{2}}}{3} = \frac{2 {\color{red}{\ln{\left(x \right)}}}^{\frac{3}{2}}}{3}$$

Portanto,

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

Adicione a constante de integração:

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

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