Question

resolva essa integral$\int\log5 {(x)} \, dx$

Answer 1

$\displaystyle \int \log_5(x)dx$
sabemos que
$\displaystyle \log_5x=\frac{\ln x}{\ln5}$
logo:
$\displaystyle \int\log_5x~dx=\int\frac{\ln x}{\ln 5}dx=\frac{1}{\ln 5}\int \ln x\,dx$
teremos que fazer o método de integração por partes:
$\displaystyle \int udv=vu-\int vdu\\\\u=\ln x\implies du=\frac{1}{x}dx\\\\v=x\implies dv=dx$
logo:
$\displaystyle i)~~~~\frac{1}{\ln 5}\int \ln xdx=\frac{1}{\ln 5}\left(x\ln x-\int dx\right)\\\\ii)~~~\frac{1}{\ln 5}\int \ln xdx=\boxed{\boxed{\frac{x\ln |x|-x}{\ln 5}+c\Longleftrightarrow \frac{\ln |x^x|-x}{\ln 5}}}$