Integration of rational functions

I) If degP ≥ degQ, we have to divide and decompose:

q(x) = quotient; r(x) = remanider

Then we have: polynomial + immediate integral or case II

II) If degP < degQ, we have three cases:

- Q only has simple real roots

- Q has multiple real roots

- Q has complex roots

II.a) Q(x) = a·(x – x₁)·(x – x₂)·…

Then, we have to look for A₁, A₂,…Є R, such that:

and we have immediate integrals (ln)

Example:

II.b) Q(x) = (x – x₁)ⁿ ·…

Then, we have to look for A₁, A₂,…,A_nЄ R, such that:

and we have immediate integrals, again

Example:

II.c) Q(x) = (ax² + bx + c)·…

Then, we have to look for M,NЄ R, such that:

and we have immediate integrals (ln + arctan)

Example:

Exercise. Solve the following integrals:

Solutions: