Question

Integration x^2+x-2/(x+1)(x+2)(x+3)

Answer 1

Step-by-step explanation:

We have,

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

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

$= \int \frac{ ( x - 1)}{(x + 1)(x + 3)} dx$

$= \int \frac{ - 1 - 1}{(x + 1)( - 1 + 3)} dx + \int \frac{ - 3 - 1}{( - 3 + 1)(x + 3)} dx$

$= \int \frac{( - 2)}{2(x + 1)} dx + \int \frac{ (- 4)}{( - 2)(x + 3)} dx$

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

$= - ln(x + 1) + 2 ln(x + 3) + c$

$= ln( \frac{(x + 3)^{2} }{(x + 1)} ) + c$