First Principles AI

Accedmychevron_right11thchevron_rightmathchevron_rightPartial Fractionschevron_rightExercise 5.2

Solution

Let:

\frac{2x^2+3x+3}{(x+1)(x^2+1)}=\frac{A}{x+1}+\frac{Bx+C}{x^2+1}

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

2x^2+3x+3=A(x^2+1)+(Bx+C)(x+1)

Solve to get:

A=1,\quad B=1,\quad C=2

Hence:

\boxed{\frac{2x^2+3x+3}{(x+1)(x^2+1)}=\frac{1}{x+1}+\frac{x+2}{x^2+1}}