Steps to find integral of 1/(x^2(x+1)(x-2)

[naoufelabs]

Hello all,

Please I want to know the steps to find the integral primitive of this function:
∫1/[(x²)*(x+1)*(x-2)] dx.

I know the result : 1/4 ln(x) - 1/3 ln(x+1) + 1/12 ln(x-2) + 1/2x

The steps for normal function like 1/[x*(x+1)*(x-2)] were like : a/x + b/x+1 + c/x-2.
My problem is that I'm confused about (x²), I see at the end like a derivative of x².
Also if we have x³, I mean (∫1/[(x³)*(x+1)*(x-2)] dx.) we will have like a + b + c + d + e. ==>

-3/8 ln(x) +1/4x² + 1/3 ln(x+1) + 1/24 ln(x-2) - 1/4x

Thank you for any help about this primitive.

 

[dextercioby]

(If this homework, it's wrongly placed. There's a special section on HW).

You may want to check if the 3 factors in the common denominator are linearly independent. Then you should assume a decomposition into 3 terms, each of them having one factor of the product in the denominator and a polynomial of one degree less in the numerator of each fraction.

So

$$\frac{Ax+B}{x^2}$$

is your first fraction in the sum of 3 and its integral is trivial, once you find out who A and B are numerically.

P.S. My name's not Kurt Lewin, there are just words attributed to him which I like and kept as a signature here.

 

[naoufelabs]

Thank you very much Mr. Kurt Lewin