List of basic formulas for integrals of fractional rational functions:
- General form of a fractional rational function:
If ( f(x) = \frac{P(x)}{Q(x)} ), where ( P(x) ) and ( Q(x) ) are polynomials, then expansion into simple fractions is often used for integration. - Integral of partial fractions:
If ( \frac{A}{(xa)^n} ), then:
[
\int \frac{A}{(xa)^n} \, dx = -\frac{A}{(n-1)(xa)^{n-1}} + C, \quad n \neq 1
]
[
\int \frac{A}{(xa)} \, dx = A \ln |xa| + C
] - Integral of a sum of fractions:
If ( f(x) = \frac{A}{xa} + \frac{B}{(xb)^n} ), then:
[
\int f(x) \, dx = A \ln |xa| + \int \frac{B}{(xb)^n} \, dx
] - Integral of a fractional rational function with a linear denominator:
If ( f(x) = \frac{Ax + B}{(xa)(xb)} ), then:
[
\int f(x) \, dx = \int \left( \frac{A}{xa} + \frac{B}{xb} \right) \, dx
] - Integral of a square denominator:
If ( f(x) = \frac{Ax + B}{(x^2 + a^2)} ), then:
[
\int f(x) \, dx = \frac{A}{2} \ln |x^2 + a^2| + \frac{B}{\sqrt{a^2}} \tan^{-1}\left(\frac{x}{a}\right) + C
] - Integral of a fractional rational function with a root:
If ( f(x) = \frac{P(x)}{\sqrt{Q(x)}} ), then the integral can be solved by substitution, depending on the form of ( Q(x) ).
These formulas can be used depending on the specific rational function you want to integrate.