Question

Calcule $\int\limits {\frac{x}{x^{2-5x+6} } } \, dx$

Answer 1

x²-5x+6

s=5

p=6

x'=2

x''=3

ax²+bx+c=a(x-x')(x-x'')

x²-5x+6=1(x-2)(x-3)

$\frac{x}{ {x}^{2} - 5x + 6 } = \frac{x}{(x - 2)(x - 3)} \\ = \frac{A}{x - 2} + \frac{B}{x - 3}$

$A = \frac{2}{ 2 - 3} = \frac{2}{ - 1} = - 2$

$B = \frac{3}{3 - 2} = \frac{3}{1} = 3$

$∫ \frac{x}{ {x}^{2} - 5x + 6 }dx \\ = ∫ \frac{ - 2}{x - 2}dx + ∫ \frac{3}{x - 3}dx$

$= - 2 ln |x - 2| +3ln |x - 3| + c \\ = ln | \frac{ {(x - 3)}^{3} }{ {(x - 2)}^{2} }| + c$