Primitive Calculus

# 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 – x

_{1})·(x – x_{2})·… Then, we have to look for A_{1}, A_{2},…Є **R**, such that:

and we have immediate integrals (ln)

Example:

II.b) Q(x) = (x – x

_{1})^{n}·… Then, we have to look for A_{1}, A_{2},…,A_{n}Є **R**, such that:

and we have immediate integrals, again

Example:

II.c) Q(x) = (ax^{2} + 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:

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