Integration by parts Theory Problem?

[MidgetDwarf]

Find the second degree polynomial P(x) that has the following properties: (a) P(0)=1, (b) P'(0)=0, (c) the indefinite integral ∫P(x)dx/(x^3(x-1)^2). Note: the the indefinite integral is a rational function. Cannot have Log terms occurring in solution.

first. I use the generic polynomial aX^2+bx+c.

When P(0)=1=C. Therefore C=1. Taking the derivative of the generic polynomial, P'(X)=2aX+B.
When P'(0)=0=B. Therefore B=0.

So far for the generic polynomial I have. P(X)= aX^2+1.

for the integral:∫(aX^2+1)dx/(X^3(x-2)^2)

breaking up the integral. ∫(aX^2)dx/(x^3(x-1)^2)+∫dx/(x^3(x-1)^2

=a∫dx/x(x-1)^2+∫dx/(x^3(x-1)^2).

The problem is. No matter how I did the the integration by parts, either choice for u. I get a ln terms for solution.

Is there something I missing?

 

[DEvens]

Yes you are missing something. You are missing the value of the constant a. That's a hint. What value of the constant a will satisfy the requirement that there are no logs in the final answer?

 

[HallsofIvy]

This not a problem in "integration by parts", it is a problem in integration by "partial fractions". Perhaps you just miswrote that. What do you get if you rewrite the integrands in terms of "partial fractions"?

 

[MidgetDwarf]

This not a problem in "integration by parts", it is a problem in integration by "partial fractions". Perhaps you just miswrote that. What do you get if you rewrite the integrands in terms of "partial fractions"?

I thought it was an integration by parts question because it appears in the integration by parts section of my book.

Upon reading your post and the previous poster, and using the method of partial fractions it became very clear. it is a=-3. If a=-3, then the ln terms "cancel", which makes all 3 conditions true.

A very fun problem.

thanks a lot.