$\int \frac{2x^7+3x^2}{x^{10}-2x^{5}+1}dx$

Question

$$\int \frac{2x^7+3x^2}{x^{10}-2x^{5}+1}dx$$

I have no idea how to approach this problem.

I just know that I have to express the top expression in some form of a derivative of the expression in the denominator. But I just couldn't figure out how.

Any help would be appreciated.

Answer 1

Rewrite the expression as

$$\int\frac{2x^7+3x^2}{(x^5-1)^2}dx = \int \frac{2x+\frac{3}{x^4}}{\left(x^2-\frac{1}{x^3}\right)^2}dx = \frac{-1}{x^2-\frac{1}{x^3}}+C $$

by dividing top and bottom by $x^6$