Question

Essa integral não consegui resolver alguém poder fazer isso passo a passo?
∫ln(5x) dx / x

Answer 1

$\displaystyle\int \dfrac{\mathrm{\ell n}(5x)}{x}\,dx\\\\\\ =\int \mathrm{\ell n\,}(5x)\cdot \dfrac{1}{x}\,dx~~~~~~\mathbf{(i)}$


Substituição:

$\mathrm{\ell n}(5x)=u~~\Rightarrow~~\dfrac{1}{5x}\cdot 5\,dx=du~~\Rightarrow~~\dfrac{1}{x}\,dx=du$


Substituindo em $\mathbf{(i)},$ a integral fica

$=\displaystyle\int u\,du\\\\\\ =\dfrac{u^{1+1}}{1+1}+C\\\\\\ =\dfrac{u^2}{2}+C\\\\\\ =\dfrac{\big(\mathrm{\ell n}(5x)\big)^2}{2}+C\\\\\\\\ \therefore~~\boxed{\begin{array}{c}\displaystyle\int \dfrac{\mathrm{\ell n}(5x)}{x}\,dx=\dfrac{\mathrm{\ell n^2\,}(5x)}{2}+C \end{array}}$


Bons estudos! :-)