Integration by Partial Fractions/Examples/Arbitrary Example 2
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Example of Use of Integration by Partial Fractions
$\ds \int \dfrac {x + 3} {x^2 + 3 x + 2} = 2 \ln \size {x + 1} - \ln \size {x + 2} + C$
Proof
\(\ds \dfrac {x + 3} {x^2 + 3 x + 2}\) | \(=\) | \(\ds \dfrac {x + 3} {\paren {x + 1} \paren {x + 2} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac 2 {x + 1} - \dfrac 1 {x + 2}\) | Partial Fractions Expansion | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \int \dfrac {x + 3} {x^2 + 3 x + 2}\) | \(=\) | \(\ds 2 \int \dfrac 1 {x + 1} - \dfrac 1 {x + 2}\) | Linear Combination of Primitives | ||||||||||
\(\ds \) | \(=\) | \(\ds 2 \ln \size {x + 1} - \ln \size {x + 2} + C\) | Primitive of $\dfrac 1 {a x + b}$ |
$\blacksquare$
Sources
- 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 by partial fractions