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

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## Example of Partial Fractions Expansion

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

## Proof

 $\ds \dfrac 1 {x \paren {x + 1}^2}$ $=$ $\ds \dfrac A x + \dfrac B {\paren {x + 1} } + \dfrac C {\paren {x + 1}^2}$ $\ds$ $=$ $\ds \dfrac {A \paren {x + 1}^2 + B x \paren {x + 1} + C x} {x \paren {x + 1}^2}$ $\ds$ $=$ $\ds \dfrac {A \paren {x^2 + 2x + 1} + B \paren {x^2 + x} + C x} {x \paren {x + 1}^2}$ $\ds$ $=$ $\ds \dfrac {\paren {A + B} x^2 + \paren {2A + B + C} x + A} {x \paren {x + 1}^2}$ $\ds A$ $=$ $\ds 1$ $\ds \paren {A + B}$ $=$ $\ds 0$ $x^2$ 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:

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

$\blacksquare$