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5uckerberg · Jun 11, 2022

Split $x^{4}$ into $x^{2}\times{x^{2}}$ .

Then we have $x^{2}\times{\left(1+\frac{3x-2}{x^{2}-3x+2}\right)}$
$x^{2}+\frac{3x^{3}-2x^{2}}{x^{2}-3x+2}$
$x^{2}+\frac{3x^{3}-9x^{2}+6+7x^{2}-6}{x^{2}-3x+2}$
$x^{2}+3x+\frac{7x^{2}-6}{x^{2}-3x+2}$
$x^{2}+3x+\frac{7x^{2}-21x+14+21x-20}{x^{2}-3x+2}$
$x^{2}+3x+7+\frac{21x-20}{x^{2}-3x+2}$
$x^{2}+3x+7+\frac{20x+x-30+10}{x^{2}-3x+2}$
$x^{2}+3x+7+\frac{20x-30}{x^{2}-3x+2}+\frac{x+10}{x^{2}-3x+2}$
$x^{2}+3x+7+\frac{20x-30}{x^{2}-3x+2}+\frac{x-\frac{3}{2}+\frac{23}{2}}{x^{2}-3x+2}$
$x^{2}+3x+7+\frac{20x-30}{x^{2}-3x+2}+\frac{x-\frac{3}{2}}{x^{2}-3x+2}+\frac{\frac{23}{2}}{x^{2}-3x+2}$

You can do the rest of this because the rest is a matter of technique

Run hard@thehsc · Jun 11, 2022

@5uckerberg can you use long division as well?

5uckerberg · Jun 11, 2022

I mean you can give that a shot as well but I am only showing one of the pathways.

stupid_girl · Jun 12, 2022

Long division followed by partial fraction decomposition is the most standard approach for integrating rational functions.

Alternatively, some people may also use partial fraction decomposition directly by noting the degree of the quotient.
Let x^4/(x^2-3x+2)=Ax^2+Bx+C+D/(x-1)+E/(x-2).

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