Question

alguem me de a resolução desse calculo por favor O VALOR DA INTEGRAL ∫_1^e▒lnx/x dx e?

Answer 1

Olá

$\displaystyle \int\limits^e_1~ { \frac{ln(x)}{x} } \, dx \\ \\ \\ \int\limits^e_1~ { \ln(x)}\cdot \frac{1}{x} } \, dx \\ \\ \\ \text{Fazendo substituicao de variavel} \\ \\ \\ u=ln(x) \\ \\ du= \frac{1}{x}dx \\ \\ \text{Substituindo na integral}$

$\displaystyle \int\limits^e_1~ { u } \, du \\ \\ \\= \left.\left(\,\mathrm{ \frac{u^2}{2} \,}\right)\right|_{1}^{e} \\ \\ \\ \text{Voltando com a variavel} \\ \\ \\ = \left.\left(\,\mathrm{ \frac{ln^2(x)}{2} \,}\right)\right|_{1}^{e} \\ \\ \\= \left.\left(\,\mathrm{ \frac{ln^2(e)}{2} \,}\right)-\left.\left(\,\mathrm{ \frac{ln^2(1)}{2} \,}\right)$

$ln(e)=1 \\ ln(1)=0 \\ \\ \\ \displaystyle =\left.\left(\,\mathrm{ \frac{1}{2} \,}\right)-0$

$\displaystyle \boxed{ =\frac{1}{2} }$