Cases of Partial Fraction Decomposition
How many cases are there in integration using partial fractions?
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$\begingroup$If I understood your question correctly, I would say there are $5$ cases.
Assume you have a rational function $\dfrac{p(x)}{q(x)}$, where the degree of $q(x)$ exceeds the degree of $p(x)$.
Case $1$: $q(x)$ is a product of distinct linear factors
Example: Consider $q(x)=\dfrac{x}{(x+3)(x-1)}$
Case $2$: $q(x)$ is a product of linear factors, where some of these factors are repeated
Example: Consider $q(x)=\dfrac{x^2}{(x+4)^2(x-2)}$
Case $3$: $q(x)$ is a product of distinct irreducible quadratic factors
Example: Consider $q(x)=\dfrac{x}{(x^2+1)(x^2+3)}$
Case $4$: $q(x)$ is a product of irreducible quadratic factors, where some are repeated
Example: Consider $q(x)=\dfrac{2x-1}{(x^2+x+1)^3}$
Case $5$: $q(x)$ is some mixture of the above cases.
Example: Consider $\dfrac{3x-2}{(x-2)^2(x^2+x+2)}$
Consider another example, which has been worked to the decomposition stage of the solution.
\begin{align} &\frac{2x-1}{(x-1)^2(x^2+x+1)^2}\\ &=\frac{A}{x-1}+\frac{B}{(x-1)^2}+\frac{Cx+D}{x^2+x+1}+\frac{Ex+F}{(x^2+x+1)^2}\\ \end{align}
Then, you can do what you normally do for partial fractions and equate the coefficients.
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