Integration of Rational Functions

Integration of Rational Functions

Every rational function can be represented in the form P(x) / Q(x) where P(x) and Q(x) are polynomials i.e., assuming of course that the polynomials do not have any common root.

If the fraction is improper, then we can always write

Just as

Few Cases

(a)

(b)

(c)

=

(d)

=

=  

(e) If 4ac – b² < 0 then (c) fails and we can reorganize.

=

(f) If the integrand is not any of the above forms we decompose the expression into partial fractions and integral separately. For example, I =

To start we change the expression to an algebraic one by putting tan x = t

we get

= ln|1 + t| = ln |1 + tanx| + c