Question

Calcule a integral indefinida ∫ x - 3

parciais.

dx usando o método das frações

x2 - 3x + 2

Answer 1

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

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

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

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

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