Partial Fraction Decomposition of 1/x^2 x^3: A Comprehensive Guide

In the realm of calculus, one of the essential operations is the decomposition of complex algebraic fractions into simpler partial fractions. This article will explore the process of decomposing the fraction 1/x^2 x^3, providing a clear, step-by-step guide that is understandable and useful for both students and professionals alike.

Understanding Partial Fraction Decomposition

Partial fraction decomposition is a technique used to express a complex rational function as the sum of simpler fractions, each with a simpler denominator. This method is particularly useful for integrating or solving differential equations involving complex rational functions.

Decomposing 1/x^2 x^3

Let's consider the rational function 1/(x^2 x^3). To decompose this function into partial fractions, we start by expressing it in the form of:

1/(x^2 x^3) A/(x^2) B/(x^3)

Here, we have chosen the form based on the fact that both denominators (x^2 and x^3) are powers of the variable x.

Step-by-Step Decomposition

First, let's rewrite the given fraction in the form of partial fractions:

Step 1: Setting Up the Equation

Starting with the equation:

1/(x^2 x^3) A/(x^2) B/(x^3)

Multiplying through by the common denominator (x^2 x^3) yields:

1 A(x^3) B(x^2)

This simplifies to:

1 Ax^3 Bx^2

Step 2: Equating Coefficients

From the equation 1 Ax^3 Bx^2, we can equate the coefficients of the powers of x on both sides. We get:

Ax^3 Bx^2 ^3 1x^2 1

From this, we obtain the system of equations:

A 0 B 1

Upon closer inspection, we realize that the above coefficients need to match, leading us to the correct values of A and B.

Step 3: Solving for A and B

By comparing the coefficients, we find:

AB 0

3A 2B 1

From AB 0, we have either A 0 or B 0. However, this does not provide a useful solution in this context. Therefore, we need to solve the system:

A 1

3(1) 2B 1

3 2B 1

2B -2

B -1

Substituting A 1 and B -1 back into the equation, we get:

1/(x^2 x^3) 1/(x^2) - 1/(x^3)

Alternative Approach

Alternatively, we can use another method to confirm the result. Using the identity 1 x^3 - x^2, we can write:

1/(x^3 x^2) (x^3 - x^2)/(x^3 x^2)

Simplifying, we get:

1/(x^3 x^2) 1/x^2 - 1/x^3

Thus, confirming our previous result.

Conclusion

In conclusion, the partial fraction decomposition of 1/(x^2 x^3) is 1/(x^2) - 1/(x^3). This method can be applied to similar problems, providing a clear and systematic approach to solving complex rational functions.

For further reading and practice, consider exploring more examples of partial fraction decomposition, integrals involving rational functions, and algebraic manipulation techniques.