The following is a list of integrals (antiderivative functions) of rational functions.
For other types of functions, see lists of integrals.
Miscellaneous integrands
Any rational function can be integrated using partial fractions in integration, by decomposing the rational function into a sum of functions of the form:
- , and
Integrands of the form xm(a x + b)n
-
- More generally,[1]
- (Cavalieri's quadrature formula)
Integrands of the form xm / (a x2 + b x + c)n
For
Integrands of the form xm (a + b xn)p
- The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m and p toward 0.
- These reduction formulas can be used for integrands having integer and/or fractional exponents.
Integrands of the form (A + B x) (a + b x)m (c + d x)n (e + f x)p
- The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m, n and p toward 0.
- These reduction formulas can be used for integrands having integer and/or fractional exponents.
- Special cases of these reductions formulas can be used for integrands of the form by setting B to 0.
Integrands of the form xm (A + B xn) (a + b xn)p (c + d xn)q
- The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m, p and q toward 0.
- These reduction formulas can be used for integrands having integer and/or fractional exponents.
- Special cases of these reductions formulas can be used for integrands of the form and by setting m and/or B to 0.
Integrands of the form (d + e x)m (a + b x + c x2)p when b2 − 4 a c = 0
- The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m and p toward 0.
- These reduction formulas can be used for integrands having integer and/or fractional exponents.
- Special cases of these reductions formulas can be used for integrands of the form when by setting m to 0.
Integrands of the form (d + e x)m (A + B x) (a + b x + c x2)p
- The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m and p toward 0.
- These reduction formulas can be used for integrands having integer and/or fractional exponents.
- Special cases of these reductions formulas can be used for integrands of the form and by setting m and/or B to 0.
Integrands of the form xm (a + b xn + c x2n)p when b2 − 4 a c = 0
- The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m and p toward 0.
- These reduction formulas can be used for integrands having integer and/or fractional exponents.
- Special cases of these reductions formulas can be used for integrands of the form when by setting m to 0.
Integrands of the form xm (A + B xn) (a + b xn + c x2n)p
- The resulting integrands are of the same form as the original integrand, so these reduction formulas can be repeatedly applied to drive the exponents m and p toward 0.
- These reduction formulas can be used for integrands having integer and/or fractional exponents.
- Special cases of these reductions formulas can be used for integrands of the form and by setting m and/or B to 0.
References
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