Question

Evaluate:

`int (2x + 1)/(x(x - 1)(x - 4)) dx`.

Answer 1

Let `I = int (2x + 1)/(x(x - 1)(x - 4)) dx`

Let `(2x + 1)/(x(x - 1)(x - 4)) = A/x + B/(x - 1) + C/(x - 4)`

∴ 2x + 1 = A(x − 1)(x − 4) + Bx(x − 4) + Cx(x − 1)     ....(i)

Putting x = 0 in (i), we get

2(0) + 1 = A(0 − 1)(0 − 4) + B(0)(0 − 4) + C(0)(0 − 1)

∴ 2(0) + 1 = A ( −1)( −4) + B0( −4) + 0( −1)

∴ 1 = 4 A

∴ A = `1/4`

Putting x − 1 = 0, i.e., x = 1 in (i), we get

2(1) + 1 = A(0)(−3) + B(1)(1 − 4) + C(1)(0)

∴ 2 + 1 = 0(−3) + B(−3) + 0

∴ 3 = −3B

∴ B = −1

Putting x − 4 = 0, i.e., x = 4 in (i), we get

2(4) + 1 = A(3)(0) + B(4)(0) + C(4)(4 − 1)

∴ 8 + 1 = 0 + 0 + 4C(3) 

∴ 9 = C(4)(3)

∴ C = `3/4`

∴ `(2x + 1)/(x(x - 1)(x - 4)) = ((1/4))/x + ((-1))/(x - 1) + ((3/4))/(x - 4)`

∴ I = `int(((1/4))/x + ((-1))/(x - 1) + ((3/4))/(x - 4)) dx` 

= `1/4 int 1/x dx - int 1/(x - 1) dx + 3/4 int 1/(x - 4) x`

∴ I = `1/4 log |x| - log |x - 1| + 3/4 log |x - 4| + c`