Chapter C, Problem 1E

To determine

Partial fraction decomposition for the given expression.

Answer to Problem 1E

Solution: The required partial fraction decomposition is:

  $\frac{5}{3x\left(2x+1\right)}=\frac{5}{3x}-\frac{10}{3\left(2x+1\right)}$

Explanation of Solution

Explanation:Given: The expression $\frac{5}{3x\left(2x+1\right)}$ .

Formula Used: Set up for partial fraction decomposition for the given expression would be of the form:

  $\frac{f\left(x\right)}{\left(x-a\right)\left(x-b\right)}=\frac{A}{\left(x-a\right)}+\frac{B}{\left(x-b\right)}$

Calculation: We set up the given expression for partial fraction decomposition as shown below:

  $\frac{5}{3x\left(2x+1\right)}=\frac{A}{x}+\frac{B}{2x+1}$

Now we need to find the values of A and B that satisfy this equation. This can be done as shown below:

  $\begin{array}{l}\Rightarrow \frac{5}{3x\left(2x+1\right)}=\frac{A}{x}+\frac{B}{2x+1}\\ \Rightarrow 5=3A\left(2x+1\right)+3Bx\\ \text{Upon comparing coefficients of all the }x\text{ terms, we get:}\\ \Rightarrow 3A=5\Rightarrow A=\frac{5}{3}\\ \Rightarrow 6A+3B=0\Rightarrow 6\left(\frac{5}{3}\right)+3B=0\Rightarrow 3B=-10\Rightarrow B=-\frac{10}{3}\\ \text{Thus, the required partial fraction decomposition is:}\\ \Rightarrow \frac{5}{3x\left(2x+1\right)}=\frac{5}{3x}-\frac{10}{3\left(2x+1\right)}\end{array}$

Conclusion: The given expression has been decomposed using partial fractions and the required decomposition has been found to be $\frac{5}{3x\left(2x+1\right)}=\frac{5}{3x}-\frac{10}{3\left(2x+1\right)}$ .