Question

Integrate the following functions with respect to x :

`(x + 1)/((x + 2)(x + 3))`

Answer 1

`(x + 1)/((x + 2)(x + 3)) = "A"/(x + 2) + "B"/(x + 3)`

Put x = – 3

– 3 + 1 = A(– 3 + 3) + B(– 3 + 2)

– 2 = A × 0 + B(– 1)

B = 2

Put x = – 2

– 2 + 1 = A(– 2 + 3) + B(– 2 + 2)

– 1 = A × 1 + B × 0

A = – 1

`(x + 1)/((x + 2)(x + 3)) = (- 1)/(x + 2) + 2/(x + 3)`

`int (x + 1)/((x + 2)(x + 3)) "d"x = int ((-1)/(x + 2) + 2/(x + 3)) "d"x`

= `- int ("d"x)/(x + 2) + 2int ("d"x)/(x + 3)`

= – log |x + 2| + 2 log |x + 3| + c

= 2 log |x + 3| – log |x + 2|+ c