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Chapter 7: Problem 40

Give the appropriate form of the partial fraction decomposition for the following functions. $$\frac{x^{2}}{x^{3}\left(x^{2}+1\right)}$$

Short Answer

Expert verified
Answer: The appropriate form of the partial fraction decomposition for the given function is: $$\frac{x^{2}}{x^{3}\left(x^{2}+1\right)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{Dx + E}{x^{2}+1}$$

Step by step solution

01

Identify the form of the denominator

First, identify all the factors of the denominator. The denominator has two factors: \(x^{3}\) and \(x^{2}+1\).
02

Determine the partial fraction decomposition form

We need to express the given function as the sum of fractions, where each of the fractions has a simpler denominator. Since we have two factors in the denominator, our partial fraction decomposition form will have two fractions: $$\frac{x^{2}}{x^{3}\left(x^{2}+1\right)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{Dx + E}{x^{2}+1}$$ In this form, A, B, C, D, and E are constants that we need to find.
03

Conclusion

The appropriate form of the partial fraction decomposition for the given function is: $$\frac{x^{2}}{x^{3}\left(x^{2}+1\right)} = \frac{A}{x} + \frac{B}{x^2} + \frac{C}{x^3} + \frac{Dx + E}{x^{2}+1}$$

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