I have this practice problem for AP Calc that I've been working on:
integral of (x^5 - 2x^2 + 1) / (x^3 - x) dx
I attempted to solve by integration by parts and am getting a different answer from the answer key.
I apologize in advance as I don't know how to format math equations and symbols on this site. I have my pictures of my work on paper and the answer key below.
I understand the process on the answer key, but I'm not sure why my process led me to a different answer. The only reason I can think of is that I somehow messed up on the process or that the fraction must be simplified before solving by integration by parts.
The answer I got derives to (2x^2 + x + 1) / (x^3 -x), the fraction portion of the answer key after rewriting with long division.
For partial fractions, degree of the numerator should be less than the degree of denominator. Observe that your solution and the book solution are almost same, there are some extra terms in the book solution that turn out to be the antiderivative of the quotient.