Question

Find:

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

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)) = 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 = A(1 + 2)(1 − 3)

⇒ 2 − 1 = A(3)(−2)

⇒ 1 = − 6A

⇒ A = `-1/6`

Put x + 2 = 0

⇒ x = −2

⇒ 2(−2) − 1 = B(−2 − 1)(−2 − 3)

⇒ −4 − 1 = B(−3)(−5)

⇒ −5 = 15B

⇒ B = `-5/15 = -1/3`

Put x − 3 = 0 

⇒ x = 3

⇒ 2(3) − 1 = C(3 − 1)(3 + 2)

⇒ 6 − 1 = (3C − C)(5)

⇒ 5 = 15C − 5C

⇒ 5 = 10C 

⇒ C = `5/10`

⇒ C = `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`