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Partial fraction decomposition process
- Check that we have a proper fraction. If not, do
long division of polynomials.
- Factor $Q(x)$ into a product of linear and
irreducible quadratic terms.
- Write $\displaystyle\frac{P(x)}{Q(x)}$ as a sum of
terms with unknown coefficients:
- For every factor of $(x-a)$ in $Q(x)$, we have
a term $\displaystyle\frac{A}{x-a}$.
- For every repeated linear factor $(x-a)^n$, we
have $$\displaystyle\frac{A_1}{(x-a)} +
\frac{A_2}{(x-a)^2} + \ldots +
\frac{A_n}{(x-a)^n}.$$
- For every irreducible quadratic factor $x^2 +
bx + c$, we have
$$\displaystyle\frac{Ax+B}{x^2+bx+c}.$$
- For every repeated irreducible quadratic factor
$(x^2+bx+c)^n$, we have $$\displaystyle\frac{A_1 x
+ B_1}{x^2+bx+c} +
\frac{A_2x+B_2}{(x^2+bx+c)^2}+\ldots +
\frac{A_nx+B_n}{(x^2+bx+c)^n}.$$
- Multiply both sides of the equation by $Q(x)$. Then
plug in different values of $x$ to get equations that
determine the coefficients. For distinct linear
factors, we can always get the coefficients by
plugging in the $x$ value that will make the factor
zero. For repeated linear factors or for quadratic
factors, it can be a little trickier.
- If all else fails, compare the coefficients of 1,
$x$, $x^2$, etc. on both sides to get equations
involving $A$, $B$, etc.
- Integrate. Recall that $\int \frac{1}{x-a}\,dx
=\ln\lvert x-a\rvert +C$, and remember substitution
(for repeated linear factors), and trig substitution
with tangent (irreducible quadratic factors).
Integrating other quadratic expressions is done with
either a $u$-substitution or a trig substitution with
tangent, after possibly completing the square.
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