Integral of $$$\frac{e \ln\left(x\right)}{x}$$$

Find $$$\int \frac{e \ln\left(x\right)}{x}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=e$$$ and $$$f{\left(x \right)} = \frac{\ln{\left(x \right)}}{x}$$$:

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

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

Then $$$du=\left(\ln{\left(x \right)}\right)^{\prime }dx = \frac{dx}{x}$$$ (steps can be seen »), and we have that $$$\frac{dx}{x} = du$$$.

Therefore,

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

Apply the power rule $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=1$$$:

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

Recall that $$$u=\ln{\left(x \right)}$$$:

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

Therefore,

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

Add the constant of integration:

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

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