Introduction to Partial Fractions

Partial fractions are a powerful tool in algebra for decomposing more complex algebraic fractions into simpler fractions. This method is particularly useful in calculus, where it simplifies the integration process. Today, we will walk through the process of decomposing a specific algebraic fraction and solve for the constants involved.

Understanding the Problem: x - 5 / (x^5 2x - 1)

Let's begin with the fraction:

$$frac{x - 5}{x^5 2x - 1}$$

We aim to decompose this into a sum of simpler fractions:

$$frac{x - 5}{x^5 2x - 1} frac{A}{x - 5} frac{B}{2x - 1}$$

Step 1: Equate and Combine Fractions

Begin by expressing the left side as a sum of fractions:

$$x - 5 A(2x - 1) B(x - 5)$$

Next, distribute (A) and (B) to obtain a common denominator:

$$x - 5 2Ax - A Bx - 5B$$

Combine like terms:

$$x - 5 (2A B)x - (A 5B)$$

Step 2: Solve for the Constants

To solve for (A) and (B), we'll set up a system of equations by comparing coefficients on both sides of the equation.

Equating the coefficients of (x), we get:

$$2A B 1$$

Equating the constants, we get:

$$-A - 5B -5$$

Solving the System of Equations

First, solve for (A) by substituting (x -5):

$$-5 - 5 A(2(-5) - 1)$$

$$-10 A(-10 - 1)$$

$$-10 A(-11)$$

$$A frac{10}{9}$$

Next, substitute (A frac{10}{9}) into the first equation to solve for (B):

$$2left(frac{10}{9}right) B 1$$

$$frac{20}{9} B 1$$

Subtract (frac{20}{9}) from both sides:

$$B 1 - frac{20}{9}$$

Convert 1 to a fraction with a denominator of 9:

$$B frac{9}{9} - frac{20}{9}$$

$$B frac{9 - 20}{9}$$

$$B -frac{11}{9}$$

Conclusion

Now we can write the partial fraction decomposition:

$$frac{x - 5}{x^5 2x - 1} frac{frac{10}{9}}{x - 5} frac{-frac{11}{9}}{2x - 1}$$

Expressing it more clearly:

$$frac{x - 5}{x^5 2x - 1} frac{10}{9(x - 5)} - frac{11}{9(2x - 1)}$$

This process can be repeated to decompose more complex algebraic fractions into simpler components, making them easier to integrate or manipulate.