Question

Calcule a integral indefinida ∫ x - 3/x2 - 3x + 2 dx usando o método das frações parciais.

Answer 1

$\int { \frac{x-3}{x^2-3x+2} } \, dx$

fatorando o denominador que tem raízes x'=1 e x''=2

$\boxed{\boxed{\int { \frac{x-3}{(x-1)(x-2)} } \, dx}}$

decompondo em frações parciais

$\frac{x-3}{(x-1)(x-2)} = \frac{A}{(x-1)}+ \frac{B}{(x-2)} \\\\ \frac{x-3}{(x-1)(x-2)} = \frac{A(x-2)+B(x-1)}{(x-1)(x-2)} \\\\ \boxed{\boxed{(x-3)=A(x-2)+B(x-1) }} \\\\ \text{quando x= 2}\\\\ (2-3)=A(2-2)+B(2-1)\\\\\boxed{-1=B}\\\\::::::::::::::\\\\ \text{quando x=1}\\\\ (1-3)=A(1-2)+B(1-1)\\\\-2=-A\\\\\boxed{2=A}\\\\\ \text{conclusao}\\\\ \boxed{\boxed{\frac{x-3}{(x-1)(x-2)} = \frac{2}{(x-1)}+ \frac{-1}{(x-2)} }}$

agora ficando com a integral:

$\int \frac{2}{(x-1)}- \frac{1}{(x-2)}\;dx\\\\ \boxed{\boxed{= \int \frac{2}{(x-1)} - \int \frac{1}{(x-2)} = 2ln(x-1)-ln(x-2)+C}}$