Integration by Partial Fractions
Theorem
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 $Q$.
Consider the primitive:
- $\ds \int \map R x \rd x$
Let $\map R x$ be expressible by the partial fractions expansion:
- $\map R x = \ds \sum_{k \mathop = 0}^n \dfrac {\map {p_k} x} {\map {q_k} x}$
where:
- each of the $\map {p_k} x$ are polynomial functions
- the degree of $\map {p_k} x$ is strictly less than the degree of $\map {q_k} x$ for all $k$.
Then:
- $\ds \int \map R x \rd x = \sum_{k \mathop = 0}^n \int \dfrac {\map {p_k} x} {\map {q_k} x} \rd x$
This technique is known as Integration by Partial Fractions.
Proof
| \(\ds \int \map R x \rd x\) | \(=\) | \(\ds \int \paren {\sum_{k \mathop = 0}^n \dfrac {\map {p_k} x} {\map {q_k} x} } \rd x\) | Definition of $\map R x$: by hypothesis | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^n \int \dfrac {\map {p_k} x} {\map {q_k} x} \rd x\) | Linear Combination of Integrals |
$\blacksquare$
Examples
Arbitrary Example $1$
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 $d$.
Arbitrary Example $2$
$\ds \int \dfrac {x + 3} {x^2 + 3 x + 2} = 2 \ln \size {x + 1} - \ln \size {x + 2} + C$
Also see
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): integration
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): integration by partial fractions
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): integration
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): integration by partial fractions