Partial Fractions

partial fractions

https://www.youtube.com/watch?v=9vll3G-LEwY integration

Algebra for polynomial fractions. This is expected knowledge.

$$\begin{array}{rl}& =\frac{x}{2x-1}-\frac{3}{x+4}\\ & \\ & =\frac{x(x+4)}{(2x-1)(x+4)}-\frac{3(2x-1)}{(2x-1)(x+4)}\\ & \\ & =\frac{x^2+4x}{2x^2+7x-4}-\frac{6x-3}{2x^2+7x-4}\\ & \\ & =\frac{x^2+4x-6x-3}{2x^2+7x-4}\end{array}$$

We can also reverse this through a process called “partial fracti ons”.

$$\begin{array}{rlr}& =\frac{x+3}{3x^2+x-10}& \\ & & \\ & =\frac{x+3}{(3x-5)(x+2)}& \\ & & \\ & =\frac{A}{3x-5}+\frac{B}{x+2}& \\ & & \\ & =\frac{A(x+2)+B(3x-5)}{(3x-5)(x+2)}& \\ & & \\ & =\frac{(A+3B)x+2A-5B}{(3x-5)(x+2)}& \\ & & \\ A+3B& =1& \\ & & \\ 2A-5B& =3& \text{We know from the first fraction}\\ & & \\ & & \\ & & \\ 2A+6B& =2& \text{multiply by 2 so things can cancel}\\ & & \\ 2A-5B& =3& \\ & & \\ \text{subtract the previous two functions}& & \\ & & \\ 0-11B& =-1& \\ & & \\ B& =-\frac{1}{11}& \\ & & \\ 3(-\frac{1}{11})+A=1& & \\ & & \\ A& =\frac{14}{11}& \\ & & \\ & & \\ & & \\ & =\frac{\frac{14}{11}}{3x-5}+\frac{-\frac{1}{11}}{x+2}& \\ & & \end{array}$$

This is useful for finding integrals of complicated fractions.

$$\begin{array}{rl}& \int \frac{x+3}{3x^2+x-10}dx\\ & \\ & =\int \frac{\frac{14}{11}}{3x-5}-\int \frac{\frac{1}{11}}{x+2}\\ & \\ & =\frac{14}{11}\int \frac{1}{3x-5}-\frac{1}{11}\int \frac{1}{x+2}\\ & \\ & =\frac{14}{33}ln|3x-5|-\frac{1}{11}ln|x+2|+C\end{array}$$

Another example

$$\begin{array}{rl}\int \frac{x-2}{x^2+5x}dx& \\ & \\ & =\frac{x-2}{x(x+5)}\\ & \\ & =\frac{A}{x}+\frac{B}{x+5}\\ & \\ & =\frac{A(x+5)}{x(x+5)}+\frac{B(x)}{x(x+5)}\\ & \\ & =\frac{(A+B)x+5A}{x(x+5)}\\ & \\ A+B& =1\\ & \\ 5A& =-2\\ & \\ A& =-\frac{2}{5}\\ & \\ B& =\frac{7}{5}\\ & \\ & =\frac{-\frac{2}{5}}{x}+\frac{\frac{7}{5}}{x+5}\\ & \\ & =\int \frac{-\frac{2}{5}}{x}+\int \frac{\frac{7}{5}}{x+5}\\ & \\ & =-\frac{2}{5}\int \frac{1}{x}+\frac{7}{5}\int \frac{1}{x+5}\\ & \\ & =-\frac{2}{5}ln|x|+\frac{7}{5}ln|x|+C\end{array}$$

Homework

Section 6.3 (pg.303): #5-12, #19-28 Section 6.4 (pg.312): #5-26, #27-32 Section 6.5 (pg.320): #7-12, 16, 17, 18, 20

6.3 7

$$\begin{array}{rl}\int \frac{7x+7}{x^2+3x-10}& \\ & \\ \frac{A}{(x-2)}+\frac{B}{(x+5)}& \\ & \\ \frac{A(x+5)}{(x-2)(x+5)}+\frac{B(x-2)}{(x-2)(x+5)}& \\ & \\ A+B=7& \\ & \\ -A2+B5=7& \\ & \\ A5+B5=35& \\ & \\ -A7=-28& \text{ subtract two funcs above}\\ & \\ A=4& \\ & \\ B=3& \\ & \\ \int \frac{4}{x-2}+\int \frac{3}{x+5}& \\ & \\ 4ln|x|+3ln|x|+C& \end{array}$$

6.3 8

$$\begin{array}{rl}\int \frac{7x-2}{x^2+x}& \\ & \\ & =\frac{A}{x}+\frac{B}{x+1}\\ & \\ A(x+1),B(x)& \\ & \\ A+B=7& \\ & \\ A=-2& \\ & \\ B=9& \\ & \\ \int \frac{-2}{x}+\frac{9}{x+1}& \\ & \\ -2ln|x|+9ln|x|& \\ & \end{array}$$