Integration by Partial Fractions/Examples/Arbitrary Example 1

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Example of Use of Integration by Partial Fractions

Let $\map R x = \dfrac {\map P x} {\map Q x}$ be a rational function over $\R$ such that the degree of the polynomial $P$ is strictly smaller than the degree of the polynomial $Q$.

Let $\map Q x$ be expressible as:

$\map Q x = \paren {x - a} \paren {x - b}^2 \paren {x^2 + c x + d}$

Then:

\(\ds \int \map R x \rd x\) \(=\) \(\ds \int \paren {\dfrac A {x - a} + \dfrac {B_1} {x - b} + \dfrac {B_2} {\paren {x - b}^2} + \dfrac {C x + D} {x^2 + c x + d} } \rd x\)
\(\ds \) \(=\) \(\ds \int \dfrac A {x - a} \rd x + \int \dfrac {B_1} {x - b} \rd x + \int \dfrac {B_2} {\paren {x - b}^2} \rd x + \int \dfrac {C x + D} {x^2 + c x + d} \rd x\)

where $A$, $B_1$, $B_2$, $C$ and $D$ are constants dependent upon $a$, $b$, $c$ and $d$.


Sources

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