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Vi3nc6en0t's question at Yahoo! Answers regarding integration by partial fraction decomposition

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MarkFL

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Feb 24, 2012
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Here is the question:

I have problems solving this(integration by partial fractions)?


∫ (x^5 + 2x^3 - 3x)dx / (x^2 + 1)^3

I really don't know. Partial fractions. I know that you're supposed to do something like Ax+B when it comes to quadratic eqn's and you're supposed to give another arbitrary constant when it's raised to a power. but I don't think I can do it unless there's something else with it and I have no idea how to factor it. I really should've listened to our earlier algebra classes. :(
I have posted a link there so the OP can view my work.
 
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MarkFL

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Feb 24, 2012
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Hello Vi3nc6en0t,

We are given to evaluate:

\(\displaystyle \int\frac{x^5+2x^3-3x}{\left(x^2+1 \right)^3}\,dx\)

First, we observe that the numerator of the integrand can be factored:

\(\displaystyle x^5+2x^3-3x=x\left(x^4+2x^2-3 \right)=x\left(x^2+3 \right)\left(x^2-1 \right)=x(x+1)(x-1)\left(x^2+3 \right)\)

Thus, there are no common factors to divide out. To complete the partial fraction decomposition, we observe that an integrand with a repeated quadratic factor will decompose as follows:

\(\displaystyle \frac{P(x)}{\left(ax^2+bx+c \right)^n}=\sum_{k=1}^n\left(\frac{A_kx+B_k}{\left(ax^2+bx+c \right)^k} \right)\)

Thus, for the given integrand, we may write:

\(\displaystyle \frac{x^5+2x^3-3x}{\left(x^2+1 \right)^3}=\frac{Ax+B}{x^2+1}+\frac{Cx+D}{\left(x^2+1 \right)^2}+\frac{Ex+F}{\left(x^2+1 \right)^3}\)

Multiplying through by \(\displaystyle \left(x^2+1 \right)^3\), we obtain:

\(\displaystyle x^5+2x^3-3x=(Ax+B)\left(x^2+1 \right)^2+(Cx+D)\left(x^2+1 \right)+(Ex+F)\)

Expanding the right side and arranging on like powers of $x$, we obtain:

\(\displaystyle x^5+2x^3-3x=Ax^5+Bx^4+(2A+C)x^3+(2B+D)x^2+(A+C+E)x+(B+D+F)\)

Equating corresponding coefficients, we obtain the system:

\(\displaystyle A=1\)

\(\displaystyle B=0\)

\(\displaystyle 2A+C=2\implies C=0\)

\(\displaystyle 2B+D=0\implies D=0\)

\(\displaystyle A+C+E=-3\implies E=-4\)

\(\displaystyle B+D+F=0\implies F=0\)

Thus, we may state:

\(\displaystyle \frac{x^5+2x^3-3x}{\left(x^2+1 \right)^3}=\frac{x}{x^2+1}-\frac{4x}{\left(x^2+1 \right)^3}\)

Now, in order to integrate, let's write:

\(\displaystyle \int\frac{x^5+2x^3-3x}{\left(x^2+1 \right)^3}\,dx=\frac{1}{2}\int\frac{2x}{x^2+1}\,dx-2\int\frac{2x}{\left(x^2+1 \right)^3}\,dx\)

For both integrals, consider the substitution:

\(\displaystyle u=x^2+1\,\therefore\,du=2x\,dx\)

And we may now write:

\(\displaystyle \int\frac{x^5+2x^3-3x}{\left(x^2+1 \right)^3}\,dx=\frac{1}{2}\int\frac{1}{u}\,du-2\int u^{-3}\,du\)

Applying the rules of integration, we obtain:

\(\displaystyle \int\frac{x^5+2x^3-3x}{\left(x^2+1 \right)^3}\,dx=\frac{1}{2}\ln|u|-2\frac{u^{-2}}{-2}+C\)

Back substituting for $u$, and applying the property of logs that a coefficient may be taken inside as an exponent we have:

\(\displaystyle \int\frac{x^5+2x^3-3x}{\left(x^2+1 \right)^3}\,dx=\ln\left(\sqrt{x^2+1} \right)+\frac{1}{\left(x^2+1 \right)^2}+C\)