AP Calculus BC / Unit 6: Integration and Accumulation of Change
6.12 Integrating Using Linear Partial Fractions
Notes
Split a rational function with distinct linear factors in the bottom:
$$\frac{1}{(x-1)(x+2)}=\frac{A}{x-1}+\frac{B}{x+2}$$
- Find $A$ and $B$ by clearing denominators and plugging in roots.
- Each piece integrates to a natural log.
Practice
$\displaystyle \frac{1}{(x-1)(x+2)}=\frac{A}{x-1}+\frac{B}{x+2}$. Then $A=$
$1=A(x+2)+B(x-1)$; at $x=1$, $1=3A$.
$\displaystyle \int\frac{dx}{x(x+1)}=$
$\displaystyle \frac{1}{x}-\frac{1}{x+1}$.
Partial fractions on the AP exam (BC) involve:
BC tests nonrepeating linear factors.
Ready for more? Take the Unit 6 test.