Partial Fractions and Improper Integrals

Denominator Factor = ax + b

A / (ax + b)

Denominator Factor = (ax + b)^n

A1 / (ax + b) + A2 / (ax + b) …

Denominator Factor = ax² + bx + c

(Ax + B) / (ax² + bx + c)

Denominator Factor = (ax² + bx + c)^n

(A1x + B1) / (ax² + bx + c) + … + (Anx + Bn) / (ax² + bx + c)^n

Rule for PFD when degree of numerator is greater than denominator

Turn into a polynomial using polynomial division and taking answer + remainder

a = - infinity / b = infinity

Replace infinity with t and evaluate limit as t approaches infinity or - infinity

Integral = a number

converges

Integral = infinity

diverges

Trick for integral (a - infinity) 1 / x^p dx

If p <= 1 then integral diverges
If p > 1 then integral converges

a / b makes integrand infinite

Take limit as t approaches a / b and replace bound with t

Integral of 1 / (x ± a)

ln | x ± a | + c