# Solving integrals using partial fractions

I'm learning partial fractions. However, I am not sure about when to put $$A, B, C..$$ into the numerator and when to put $$Ax+B, Bx+C..$$ into the numerator.

For example, I have the following integral:

$$\int \frac{x^5}{(x-1)^2(x^2-1)}dx$$

and I want to calculate simplify using partial fractions.
I know that $$(x-1)^2$$ has just one real solution, so I will write it as $$\frac{A}{x-1}+\frac{B}{(x-1)^2}$$.
However, $$x^2-1$$ has two real solutions, $$\{-1, 1\}$$.
Do I write is as $$\frac{Cx+D}{(x^2-1)}$$ or just $$\frac{C}{(x^2-1)}$$?

I am really not sure about when to use the simple form $$A,B$$ and when to use $$Cx+D, Dx+E$$.

Thanks

• $(x-1)(x^2-1)=(x-1)^3(x+1)$ Jan 14, 2020 at 9:42
• @IsaacYIUMathStudio Thanks, but I rather need some theory about how to use partial fractions. Jan 14, 2020 at 9:44

Actually, since the degree of the numerator is not smaller than the degree of the denominator, your first step should be to write $$x^5$$ as$$(x+2)\times\bigl((x-1)^2(x^2-1)\bigr)+4x^3-2x^2-3x+2.$$So$$\frac{x^5}{(x-1)^2(x^2-1)}=x+2+\frac{4x^3-2x^2-3 x+2}{(x-1)^2(x^2-1)}.$$On the other hand,$$x^2-1=(x-1)(x+1)\implies(x-1)^2(x^2-1)=(x-1)^3(x+1)$$and therefore you should try to get $$A$$, $$B$$, $$C$$ and $$D$$ such that$$\frac{4x^3-2x^2-3 x+2}{(x-1)^2(x^2-1)}=\frac A{x-1}+\frac B{(x-1)^2}+\frac C{(x-1)^3}+\frac D{x+1}.$$

• So the nominator can't have higher degree than the denominator? Jan 14, 2020 at 9:54
• No, it cannot. ${}$ Jan 14, 2020 at 10:04
• But in school we solved $\frac{2x^3+8x^2-8x-22}{(x+3)^2(x^2+1)}$ and we went straight to partial fractions, we didn't lower the degree of denominator. Could you explain why? And also I use the $Ax+B$ form only when I have no real solutions or also when we have two real solutions? Jan 14, 2020 at 10:09
• In that example, the numerator has degree $3$ and the denominator has degree $4$. So, in that case you are ready to do the partial fraction decomposition. Jan 14, 2020 at 10:11

A general rule of thumb is to never leave things that can be further simplified. For example,

• prefer $$\frac{A}{x+1} + \frac{B}{x-1}$$ to $$\frac{Cx+D}{x^2-1}$$

• prefer $$\frac{A}{x}+\frac{B}{x^2}$$ to $$\frac{Ax+B}{x^2}$$

Of course, you are not obliged to follow this rule to the point - if you see a clear gain to not use, do not use it.

• Okay, but could you please answer my question - do we use the form $Ax+B$ when we have no real solutions and when we have two real solutions? Jan 14, 2020 at 10:10