# Math Insight

### Partial fractions

Now we return to a more special but still important technique of doing indefinite integrals. This depends on a good trick from algebra to transform complicated rational functions into simpler ones. Rather than try to formally describe the general fact, we'll do the two simplest families of examples.

Consider the integral $$\int {1\over x(x-1)}\; dx$$ As it stands, we do not recognize this as the derivative of anything. However, we have $${1\over x-1}-{1\over x}={x-(x-1)\over x(x-1)}={1\over x(x-1)}$$ Therefore, $$\int {1\over x(x-1)}\; dx=\int {1\over x-1}-{1\over x}\;dx=\ln(x-1)-\ln x + C$$ That is, by separating the fraction $1/x(x-1)$ into the ‘partial’ fractions $1/x$ and $1/(x-1)$ we were able to do the integrals immediately by using the logarithm. How to see such identities?

Well, let's look at a situation $${cx+d\over (x-a)(x-b)}={A\over x-a}+{B\over x-b}$$ where $a,b$ are given numbers (not equal) and we are to find $A,B$ which make this true. If we can find the $A,B$ then we can integrate $(cx+d)/(x-a)(x-b)$ simply by using logarithms: $$\int {cx+d\over (x-a)(x-b)}\;dx=\int{A\over x-a}+{B\over x-b}\;dx= A\ln(x-a)+B\ln(x-b)+C$$ To find the $A,B$, multiply through by $(x-a)(x-b)$ to get $$cx+d=A(x-b)+B(x-a)$$ When $x=a$ the $x-a$ factor is $0$, so this equation becomes $$c\cdot a+d=A(a-b)$$ Likewise, when $x=b$ the $x-b$ factor is $0$, so we also have $$c\cdot b+d=B(b-a)$$ That is, $$A={c\cdot a+d\over a-b}\;\;\;\;\;B={c\cdot b+d\over b-a}$$ So, yes, we can find the constants to break the fraction $(cx+d)/(x-a)(x-b)$ down into simpler ‘partial’ fractions.

Further, if the numerator is of bigger degree than $1$, then before executing the previous algebra trick we must first divide the numerator by the denominator to get a remainder of smaller degree. A simple example is $${x^3+4x^2-x+1\over x(x-1)}=?$$ We must recall how to divide polynomials by polynomials and get a remainder of lower degree than the divisor. Here we would divide the $x^3+4x^2-x+1$ by $x(x-1)=x^2-x$ to get a remainder of degree less than $2$ (the degree of $x^2-x$). We would obtain $${x^3+4x^2-x+1\over x(x-1)}=x+5+{4x+1\over x(x-1)}$$ since the quotient is $x+5$ and the remainder is $4x+1$. Thus, in this situation $$\int {x^3+4x^2-x+1\over x(x-1)}\;dx=\int x+5+{4x+1\over x(x-1)}\;dx$$ Now we are ready to continue with the first algebra trick.

In this case, the first trick is applied to $${4x+1\over x(x-1)}$$ We want constants $A,B$ so that $${4x+1\over x(x-1)}={A\over x}+{B\over x-1}$$ As above, multiply through by $x(x-1)$ to get $$4x+1=A(x-1)+Bx$$ and plug in the two values $0,1$ to get $$4\cdot 0+1=-A\;\;\;\;\;4\cdot 1+1=B$$ That is, $A=-1$ and $B=5$.

Putting this together, we have $${x^3+4x^2-x+1\over x(x-1)}=x+5+{-1\over x}+{5\over x-1}$$ Thus, $$\int {x^3+4x^2-x+1\over x(x-1)}\;dx=\int x+5+{-1\over x}+{5\over x-1}\;dx$$ $$={x^2\over 2}+5x-\ln x+5\ln(x-1)+C$$

In a slightly different direction: we can do any integral of the form $$\int {ax+b\over 1+x^2}\;dx$$ because we know two different sorts of integrals with that same denominator: $$\int{1\over 1+x^2}\;dx=\arctan x+C\;\;\;\;\; \int {2x\over 1+x^2}\;dx=\ln(1+x^2)+C$$ where in the second one we use a substitution. Thus, we have to break the given integral into two parts to do it: $$\int {ax+b\over 1+x^2}\;dx={a\over 2}\int {2x\over 1+x^2}\;dx +b\int{1\over 1+x^2}\;dx$$ $$={a\over 2}\ln(1+x^2)+b\arctan x+C$$

And, as in the first example, if we are given a numerator of degree $2$ or larger, then we divide first, to get a remainder of lower degree. For example, in the case of $$\int {x^4+2x^3+x^2+3x+1\over 1+x^2}\;dx$$ we divide the numerator by the denominator, to allow us to write $${x^4+2x^3+x^2+3x+1\over 1+x^2}=x^2+2x+{x+1\over 1+x^2}$$ since the quotient is $x^2+2x$ and the remainder is $x+1$. Then $$\int {x^4+2x^3+x^2+3x+1\over 1+x^2}\;dx= \int x^2+2x+{x+1\over 1+x^2}$$ $$={x^3\over 3}+x^2+{1\over 2}\ln(1+x^2)+\arctan x+C$$

These two examples are just the simplest, but illustrate the idea of using algebra to simplify rational functions.

#### Exercises

1. $\int { 1 \over x(x-1) }\,dx=?$
2. $\int { 1+x \over 1+x^2 }\,dx=?$
3. $\int { 2x^3+4 \over x(x+1) }\,dx=?$
4. $\int { 2+2x+x^2 \over 1+x^2 }\,dx=?$
5. $\int { 2x^3+4 \over x^2-1 }\,dx=?$
6. $\int { 2+3x \over 1+x^2 }\,dx=?$
7. $\int { x^3+1 \over (x-1)(x-2) }\,dx=?$
8. $\int { x^3+1 \over x^2+1 }\,dx=?$