Question

o valor da integral ʃ1 e. lnx/x dx

Answer 1

$\displaystyle\int_{1}^{e}\dfrac{\ln x}{x}\,dx=\int_{1}^{e}\ln x\cdot\frac{1}{x}\,dx$

Fazendo $u=\ln x~~~\Longrightarrow~~~du=\frac{1}{x}\,dx$, temos

$\bullet\,\,x=1~~~\Longrightarrow~~~u=\ln1=0\\\\\bullet\,\,x=e~~~\Longrightarrow~~~u=\ln e=1$

Portanto, fazendo a substituição, temos que

$\displaystyle\int_{1}^{e}\dfrac{\ln x}{x}dx=\int_{1}^{e}\ln x\,\frac{1}{x}\,dx=\int_{0}^{1}u\,du=\bigg[\dfrac{u^{2}}{2}\bigg]_{u=0}^{u=1}$

Então:

$\displaystyle\int_{1}^{e}\dfrac{\ln x}{x}\,dx=\dfrac{1^{2}}{2}-\dfrac{0^{2}}{2}\\\\\\\boxed{\boxed{\int_{1}^{e}\dfrac{\ln x}{x}\,dx=\dfrac{1}{2}}}$