Question

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

Answer 1

$\int\ \frac{(x-3)}{x^2-3x+2)}\ dx\\\\ \int\ \frac{x-3}{(x-2)(x-1)}\ dx=\int\ \frac{A(x-1)+B(x-2)}{(x-2)(x-1)}\ dx$

Cancelando os termos:

$x-3=A(x-1)+B(x-2)\\\\ x=1\\\\ 1-3=0A+B(1-2)\\\\ -2=-B\\\\ \boxed{B=2}\\ \\ x=2\\\\ 2-3=A(2-1)+0B\\\\ -1=A\\\\ \boxed{A=-1}$

Então:

$\int\ \frac{(x-3)}{x^2-3x+2}\ dx=-\int \frac{1}{x-2}\ dx+2\int\ \frac{1}{x-1}\ dx\\\\ \large\boxed{\int=-ln\ |x+2|+2\ ln|x+1|+C}$