Question

DESESPERADO

Determine a regiao limitada pela curva y=lnx, pelo eixo x e pelas retas x=1 e x=e

Answer 1

$\Bmatrix{y=ln(x)\\\\x=1\\x=e\end$

como 
e ≈ 2.71828 ,

area limitada por essas curvas

$\int\limits^e_1 {ln(x)} \, dx$

integrando por partes
$\boxed{\int U.dV = U*V-\int .VdU}$

então
$U = ln(x)\\\\dU= \frac{1}{x}.dx\\\\dV=1.dx\\\\V=\int 1dx = x$

colocando na formula da integral por partes
$ln(x)*x -\int x* \frac{1}{x} dx\\\\=ln(x)*x-\int 1dx\\\\=ln(x)*x-x+C$

logo

$\boxed{\int ln(x) dx=ln(x)*x-x+C}$

mas como vamos calcular a integral definida ..fica

$\int\limits^e_1 {ln(x)} \, dx\\\\ =\left ln(x)*x-x \right |^e_1\\\\= [ln(e)*e-e ]- [ln(1)*1-1]\\\\=[1*e -e]-[0*1-1]\\\\=0 - [-1]\\\\=1\\\\\\\\\boxed{\boxed{ \int\limits^e_1 {ln(x)} \, dx=1}}$