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Partial Fraction Decomposition. Show the form of the partial fraction decomposition for the...

Question:

Partial Fraction Decomposition. Show the form of the partial fraction decomposition for the integrand below. Generate a system of equations that could be used to complete the decomposition. Rather than continuing the partial fraction decomposition, evaluate the integral using the substitution {eq}u^{2}=1+x^{2} {/eq}

{eq}\int_{0}^{\infty} \frac{x^{3}}{\left(1+x^{2}\right)^{3}} d x {/eq}

Partial Fraction Decomposition:

When you have a rational function {eq}\frac{f(x)}{g(x)} {/eq}, you can rewrite the function as a sum of simpler functions by using the partial fraction decomposition (also called partial fraction expansion). This is the same as the procedure of adding fractions by first finding a common denominator, but done in reverse order. Rewriting an integrand using partial fraction decomposition can result in integrals that are simpler to evaluate. In order to find the partial fraction decomposition, you will need to remember factoring skills and how to solve a system of equations

Answer and Explanation:

To find the form of the partial fraction decomposition of the integrand of {eq}\int_{0}^{\infty} \frac{x^{3}}{\left(1+x^{2}\right)^{3}} d x {/eq}, begin by just looking at the integrand {eq}\frac{x^{3}}{\left(1+x^{2}\right)^{3}} {/eq}.

The degree of the numerator is 3, and the degree of the denominator is 6, so since the degree of the numerator is less than the degree of the denominator, we do not need to begin with polynomial division. Instead, make sure the denominator is completely factored - in this example, the denominator is already factored into an irreducible quadratic to the third power.


Now we set up the partial fraction decomposition. When you have an irreducible quadratic as a factor, the corresponding term in the decomposition will have something similar to {eq}Ax+B {/eq} as the numerator. In this problem, we have repeated irreducible quadratic factors, so we have one term in the decomposition for each power of the factor.


{eq}\frac{x^{3}}{\left(1+x^{2}\right)^{3}} = \frac{Ax+B}{1+x^2} + \frac{Cx+D}{(1+x^2)^2} + \frac{Ex+F}{(1+x^2)^3} {/eq}


This is the form of the partial fraction decomposition. In order to find the constants {eq}A,B,C,D,E,F {/eq}, we can set up a system of equations. Begin by multiplying both sides of the equation by the factored denominator and simplifying by canceling like factors.


{eq}x^3=(Ax+B)(1+x^2)^2 + (Cx+D)(1+x^2)+(Ex+F) {/eq}


Now distribute carefully and combine like terms on the right side of the equation


{eq}x^3 = (Ax+B)(1+2x^2+x^4) + Cx + Cx^3 + D + Dx^2 + Ex + F \\ \\ x^3 = Ax + 2Ax^2 + Ax^5 + B + 2Bx^2 + Bx^4 + Cx + Cx^3 + D + Dx^2 + Ex +F\\ \\ x^3 = Ax^5 + Bx^4 + Cx^3 + (2A+2B+D)x^2 + (A+C+E)x + (B+D+E) {/eq}


Now to find the constants, we compare coefficients on each side of the equation to set up a system of equations. The coefficient of each power of x on the left side must equal the coefficient of the same power of x on the right side. So we have the system of equations


{eq}0=A\\ 0=B\\ 1=C\\ 0=2A+2B+D\\ 0=A+C+E\\ 0=B+D+F {/eq}

If we were to continue using partial fraction decomposition to evaluate this integral, we would need to solve this system to find our constants. However, this problem wants us to use the substitution {eq}u^{2}=1+x^{2} {/eq} rather than use partial fractions.


First, notice that this is an improper integral. We must rewrite the integral with a limit.

{eq}\int_{0}^{\infty} \frac{x^{3}}{\left(1+x^{2}\right)^{3}} d x = \lim_{t\rightarrow \infty} \int_{0}^{t} \frac{x^{3}}{\left(1+x^{2}\right)^{3}} d x {/eq}

Now use substitution. Let {eq}u^{2}=1+x^{2}\\ \\ u=\sqrt{1+x^2}\\ \\ du=\frac{x}{\sqrt{1+x^2}} dx\\ \\ x^2=u^2-1 {/eq}


Let's find the antiderivative using this substitution before dealing with the bounds of integration and the limit.


{eq}\int \frac{x^{3}}{\left(1+x^{2}\right)^{3}} d x = \int \frac{u^2-1}{u^5} d u = \int (u^{-3}-u^{-5}) d u\\ =-\frac{1}{2u^2}+\frac{1}{4u^4} +c = -\frac{1}{2(1+x^2)}+\frac{1}{4(1+x^2)^2}+c {/eq}


And so

{eq}\lim_{t\rightarrow \infty} \int_{0}^{t} \frac{x^{3}}{\left(1+x^{2}\right)^{3}} d x = \lim_{t\rightarrow\infty} ( -\frac{1}{2(1+t^2)}+\frac{1}{4(1+t^2)^2} +\frac{1}{2}-\frac{1}{4})=\frac{1}{4} {/eq}


Learn more about this topic:

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Partial Fraction Decomposition: Rules & Examples

from High School Algebra I: Help and Review

Chapter 3 / Lesson 25
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