Partial Fraction Decomposition

A Rational Function $P(x)/Q(x)$ can be rewritten using what is known as partial fraction decomposition. This procedure often allows integration to be performed on each term separately by inspection. For each factor of $Q(x)$ the form $(ax+b)^m$, introduce terms

$${A_1\over ax+b} + {A_2\over (ax+b)^2} + \ldots + {A_m\over (ax+b)^m}.$$ (1)

For each factor of the form $(ax^2+bx+c)^m$, introduce terms

$${A_1x+B_1\over ax^2+bx+c} + {A_2x+B_2\over (ax^2+bx+c)^2} + \ldots + {A_mx+B_m\over (ax^2+bx+c)^m}.$$ (2)

Then write

$${P(x)\over Q(x)} = {A_1\over ax+b} + \ldots + {A_2x+B_2\over ax^2+bx+c} +\ldots$$ (3)

and solve for the $A_i$s and $B_i$s.

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 13-15, 1987.