Ready to unlock your full learning experience? Start today - and unlock your math superpower! Finally master math - for real Full access to the entire curriculum Video tutorials that actually work The fastest way to get a better math grade Unlock now How to Use Partial Fraction Decomposition for Integration Partial fraction decomposition has a long name, but it is an easy and simple method. You use the factors in the denominator to create new and nicer-looking fractions. When the new fractions are set, the integration becomes much easier. Nice!
Instructions for Partial Fraction Decomposition 1. Factorize the denominator by finding the zeros . 2. Intermediate calculation (see the box below): a) Set the expression equal to a sum of fractions and choose the constants A , B , C , … b) Multiply by the common denominator. c) Sort the different terms individually. d) Create a system of equations and solve for A , B , C , …
3. Insert the result into the integral and integrate. Yay!
Compute ∫ 3 x x 2 − x − 2 d x
∫ 3 x x 2 − x − 2 d x = ∫ 3 x ( x − 2 ) ( x + 1 ) d x = ∗ ∫ 2 x − 2 + 1 x + 1 d x = 2 ln | x − 2 | + ln | x + 1 | + C
∫ 3 x x 2 − x − 2 d x = ∫ 3 x ( x − 2 ) ( x + 1 ) d x = ∗ ∫ 2 x − 2 + 1 x + 1 d x = 2 ln | x − 2 | + ln | x + 1 | + C
3 x ( x − 2 ) ( x + 1 ) = A x − 2 + B x + 1 3 x = A ( x + 1 ) + B ( x − 2 ) 3 x = A x + A + B x − 2 B 3 x + 0 = ( A + B ) x + ( A − 2 B ) Equate terms of the same degree and solve. You then get A + B = 3 and A − 2 B = 0 , which gives A = 2 and B = 1 .
3 x ( x − 2 ) ( x + 1 ) = A x − 2 + B x + 1 3 x = A ( x + 1 ) + B ( x − 2 ) 3 x = A x + A + B x − 2 B 3 x + 0 = ( A + B ) x + ( A − 2 B ) Equate terms of the same degree and solve. You then get A + B = 3 and A − 2 B = 0 , which gives A = 2 and B = 1 .
