Break a complicated fraction into a sum of simpler ones — useful for integrating in calculus.
Walk through
Step 1 of 4
Set up the form
Write 1/((x − 1)(x + 2)) = A/(x − 1) + B/(x + 2).
Choosing the right form
- Distinct linear factors (x − a)(x − b): use A/(x − a) + B/(x − b).
- Repeated factor (x − a)²: use A/(x − a) + B/(x − a)².
- Irreducible quadratic (x² + 1): use (Ax + B)/(x² + 1).
- Always make sure the top degree is smaller than the bottom first — divide if not.
Your turn
Split 5/((x)(x − 5)) into partial fractions.
Partial fractions is the standard trick for integrating rational functions in calculus — each simple piece integrates to a log or an arctan.
Try it
1/((x−1)(x+2)) = A/(x−1) + B/(x+2)
Solve for A and B: A = 1/3, B = −1/3.