Explicação passo-a-passo:
Integral Definido
Dado o integral [tex]\displaystyle\int\limits^{e/3}_{1/3}\dfrac{6\ln(3x)}{x}dx \\[/tex]
Seja [tex]I~=~\displaystyle\int\limits^{e/3}_{1/3}\dfrac{6\ln(3x)}{x}dx \\[/tex]
[tex]I~=~2\displaystyle\int\limits^{e/3}_{1/3}\ln(3x)d\left[\ln(3x)\right] \\[/tex]
[tex]I~=~ \ln^2(3x)\Big|^{e/3}_{1/3}=\ln^2\left(3*\dfrac{e}{3}\right)-\ln^2\left(3*\dfrac{1}{3}\right) \\[/tex]
[tex]I~=~\ln^2(e)-\ln^2(1)=1-0 \\[/tex]
[tex]I~=~\boxed{ 1 } \\[/tex]
