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I have to integrate the following and I was having trouble with it because I am pretty new to the concept of using partial fractions. I did as much as I could and I need help in moving forward:

$$\int\frac{x^5+3x^2+1}{x^5+x^2}dx$$

I did some long division and got:

$$\int1+\frac{2x^2+1}{x^5+x^2}dx$$

Factored the denominator:

$$x^5+x^2$$

and got the following:

$$x^2(x+1)(x^2-x+1)$$

Now I think that I should have something like this, but I'm not sure:

$$\frac{A}{x^2}+\frac{B}{x+1}+\frac{C}{x^2-x+1}$$

Would appreciate any help. Thanks.

  • 1
    Your error is only in the set-up. Because of the quadratic denominator in the first partial fraction, you need a *linear* numerator, $\frac{A+Dx}{x^2}$. Alternatively, use the set-up given by Pete Clark.2012-03-02
  • 0
    I think the above comment is most helpful. In general, should the numerator be a polynomial of one degree lower than the denominator? Should we have Ax+B/(x^2) + C/(x+1) + Dx+E/(x^2-x+1)?2012-07-16

2 Answers 2