First Principles AI

Accedmychevron_right11thchevron_rightmathchevron_rightPartial Fractionschevron_rightExercise 5.1

Solution

Let:

\frac{x+1}{(x-1)^2}=\frac{A}{x-1}+\frac{B}{(x-1)^2}

Multiply by $(x-1)^2$ :

x+1=A(x-1)+B

Put $x=1$ :

2=B

Compare coefficients of $x$ :

1=A

Hence:

\boxed{\frac{x+1}{(x-1)^2}=\frac{1}{x-1}+\frac{2}{(x-1)^2}}