Integration by Partial Fractions/Examples/Arbitrary Example 2

Example of Use of Integration by Partial Fractions

$\ds \int \dfrac {x + 3} {x^2 + 3 x + 2} = 2 \ln \size {x + 1} - \ln \size {x + 2} + C$

$\ds \dfrac {x + 3} {x^2 + 3 x + 2}$

$=$

$\ds \dfrac {x + 3} {\paren {x + 1} \paren {x + 2} }$

$\ds $

$=$

$\ds \dfrac 2 {x + 1} - \dfrac 1 {x + 2}$

$\ds \leadsto \ \ $

$\ds \int \dfrac {x + 3} {x^2 + 3 x + 2}$

$=$

$\ds 2 \int \dfrac 1 {x + 1} - \dfrac 1 {x + 2}$

$\ds $

$=$

$\ds 2 \ln \size {x + 1} - \ln \size {x + 2} + C$

$\blacksquare$