Question

Solve the intergal....... (2x-1)/[x(x-2)(x-3)]........ Please help fast?

Answer 1

`5/3ln|(x-3)|-3/2ln|(x-2)|-1/6ln|x|+C, or,`

`ln|(x-3)^(5/3)/{(x-2)^(3/2)x^(1/6)}|+C`.

Explanation:

Suppose that, `I=int(2x-1)/{x(x-2)(x-3)}dx`.

`:. I=int{(2x)/{x(x-2)(x-3)}+(-1)/{x(x-2)(x-3)}}dx`,

`=int{2/{(x-2)(x-3)}+{(x-3)-(x-2)}/{x(x-2)(x-3)}}dx`,

`=int{2/{(x-2)(x-3)}+(x-3)/{x(x-2)(x-3)}-(x-2)/{x(x-2)(x-3)}}dx`,

`=2int1/{(x-2)(x-3)}dx+int1/{x(x-2)}dx-1/{x(x-3)}dx`,

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

`-1/3int{x-(x-3)}/{x(x-3)}dx`,

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

`-1/3int{1/(x-3)-1/x}dx`,

`=2ln|(x-3)|-2ln|(x-2)|+1/2ln|(x-2)|-1/2ln|x|-1/3ln|(x-3)|+1/3ln|x|`,

`=(2-1/3)ln|(x-3)|-(2-1/2)ln|(x-2)|-(1/2-1/3)ln|x|`.

`rArr I=5/3ln|(x-3)|-3/2ln|(x-2)|-1/6ln|x|+C, or,`

`I=ln|(x-3)^(5/3)/{(x-2)^(3/2)x^(1/6)}|+C`.

Answer 2

`5/3ln|x-3|-3/2*ln|x-2|-1/6*ln|x|+C`

Explanation:

Converting your Integrand in partial fractions we get

`(2x-1)/(x*(x-2)(x-3))=5/(3*(x-3))-3/(2*(x-2))-1/(6*x)`

The result is to be seen above.