Question

Find:

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

Answer 1

I = `int(2x - 1)/((x - 1)(x + 2)(x - 3))  dx` by partial fraction

`int(2x - 1)/((x - 1)(x + 2)(x - 3))  dx = A/((x - 1)) + B/((x + 2))  + C/((x - 3))`

 2x − 1 = A(x + 2)(x − 3) + B(x − 1)(x − 3) + C(x − 1)(x + 2)

Put x − 1 = 0

x = 1

2(1) − 1 − 1 = A(1 + 2)(1 − 3)

1 = −6A
A = `-1/6`

Put x + 2 = 0

x = −2

2(−2) − 1 = B(−2 − 1)(−2 − 3)

−5 = 15B

B = `-5/15`

= `-1/3`

Put x − 3 = 0

x = 3

2(3) − 1 = C(3 − 1)(3 + 2)

5 = 10C

C = `5/10`

= `1/2`

I = `(-1)/6int1/(x - 1)  dx - 1/3int1/(x + 2)  dx + 1/2int1/(x - 3)  dx`

I = `(-1)/6 log |x - 1| - 1/3 log |x + 2| + 1/2 log |x + 3| + C`