Evaluate: integral of ln(5x) x

Expression: $\int{ \ln\left({5x}\right) } \mathrm{d} x$

Use $\ln\left({x \times y}\right)=\ln\left({x}\right)+\ln\left({y}\right)$ to expand the expression

$\int{ \ln\left({5}\right)+\ln\left({x}\right) } \mathrm{d} x$

Use the property of integral $\int{ f\left( x \right)\pmg\left( x \right) } \mathrm{d} x=\int{ f\left( x \right) } \mathrm{d} x\pm\int{ g\left( x \right) } \mathrm{d} x$

$\int{ \ln\left({5}\right) } \mathrm{d} x+\int{ \ln\left({x}\right) } \mathrm{d} x$

Use $\int{ a } \mathrm{d} x=a \times x$ to evaluate the integral

$\ln\left({5}\right)x+\int{ \ln\left({x}\right) } \mathrm{d} x$

Evaluate the indefinite integral

$\ln\left({5}\right)x+\ln\left({x}\right) \times x-x$

Add the constant of integration $C \in ℝ$

$\begin{array} { l }\ln\left({5}\right)x+\ln\left({x}\right) \times x-x+C,& C \in ℝ\end{array}$