List of integrals of rational functions

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The following is a list of integrals (antiderivative functions) of rational functions. Any rational function can be integrated by partial fraction decomposition of the function into a sum of functions of the form:

, and

which can then be integrated term by term.

For other types of functions, see lists of integrals.

Miscellaneous integrands[edit]

Integrands of the form xm(a x + b)n[edit]

Many of the following antiderivatives have a term of the form ln |ax + b|. Because this is undefined when x = −b / a, the most general form of the antiderivative replaces the constant of integration with a locally constant function.[1] However, it is conventional to omit this from the notation. For example,

is usually abbreviated as

where C is to be understood as notation for a locally constant function of x. This convention will be adhered to in the following.

(Cavalieri's quadrature formula)

Integrands of the form xm / (a x2 + b x + c)n[edit]

For



Integrands of the form xm (a + b xn)p[edit]

  • 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[edit]

  • 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[edit]

  • 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[edit]

  • 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[edit]

  • 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[edit]

  • 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[edit]

  • 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[edit]

  1. ^ "Reader Survey: log|x| + C", Tom Leinster, The n-category Café, March 19, 2012