Partial Fractions

Had to remember how to do it for MATH213.

For example Evaluate

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

Solution: For integrals of this form we use partial fractions to simplify the integrand. We first need to find an appropriate form for this problem is

$$\frac{x+2}{x^2(x+1)}=\frac{A}{x^2}+\frac{B}{x}+\frac{C}{x+1}$$

We now need to find A, B and C. There are a few standard options for how to do this:

• Multiply by $x^2(x+1)$, set the coefficients of the polynomials equal to each other and finally solve the resulting linear system for A,B,C. This is generally the last resort option.
• Pick three different values for x and then solve the linear system. This works but still involves solving a MATH 115 problem.
• Try to carefully manipulate the expression and selectively substitute in values for $x$ to quickly isolate for the coefficients. If there are no repeated roots, then this method will always work and we can simply remove the various terms in the denominator and then plug in the root of the polynomial in for $x$. For repeated roots you will generally need to solve a system of equations but can reduce the complexity of the system if you try to first find all the coefficients for the higher power terms. See the wiki article for the Heaviside cover-up method for more details

We choose option 3 as it is almost always the best for manual computations. multiplying by $x^2$ gives

$$\frac{x+2}{x+1}=A+Bx+\frac{C{x}^2}{x+1}$$

Plugging in $x=0$ gives

$$2=A$$

Next we multiply by $x+1$ to get

$$\frac{x+2}{x^2}=(x+1)(\frac{A}{x^2}+\frac{B}{x})+C$$

plugging in $x=-1$ gives

$$1=C$$

todo To finish writing