Partial Fractions Expansion/Examples/Reciprocal of x by x + 1 squared

Example of Partial Fractions Expansion

$\dfrac 1 {x \paren {x + 1}^2} = \dfrac 1 x - \dfrac 1 {\paren {x + 1} } - \dfrac 1 {\paren {x + 1}^2}$

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

$=$

$\ds \dfrac A x + \dfrac B {\paren {x + 1} } + \dfrac C {\paren {x + 1}^2}$

$\ds $

$=$

$\ds \dfrac {A \paren {x + 1}^2 + B x \paren {x + 1} + C x} {x \paren {x + 1}^2}$

$\ds $

$=$

$\ds \dfrac {A \paren {x^2 + 2x + 1} + B \paren {x^2 + x} + C x} {x \paren {x + 1}^2}$

$\ds $

$=$

$\ds \dfrac {\paren {A + B} x^2 + \paren {2A + B + C} x + A} {x \paren {x + 1}^2}$

$\ds A$

$=$

$\ds 1$

$\ds \paren {A + B}$

$=$

$\ds 0$

$x^2$ term vanishes

$\ds \leadsto \ \ $

$\ds B$

$=$

$\ds -1$

$\ds \paren {2A + B + C}$

$=$

$\ds 0$

$x$ term vanishes

$\ds \leadsto \ \ $

$\ds C$

$=$

$\ds -1$

Therefore:

$\dfrac 1 {x \paren {x + 1}^2} = \dfrac 1 x - \dfrac 1 {\paren {x + 1} } - \dfrac 1 {\paren {x + 1}^2}$

$\blacksquare$