The following is a list of
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
s (
antiderivative
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
functions) of
rational function
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
s.
Any rational function can be integrated by
partial fraction decomposition
In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction as ...
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
Integration is the basic operation in integral calculus. While differentiation has straightforward rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, ...
.
Miscellaneous integrands
:
:
:
:
:
Integrands of the form ''x''''m''(''a x'' + ''b'')''n''
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
In calculus, the constant of integration, often denoted by C (or c), is a constant term added to an antiderivative of a function f(x) to indicate that the indefinite integral of f(x) (i.e., the set of all antiderivatives of f(x)), on a connected ...
with a
locally constant function
In mathematics, a locally constant function is a function from a topological space into a set with the property that around every point of its domain, there exists some neighborhood of that point on which it restricts to a constant function.
...
.
[Reader Survey: log, ''x'', + ''C'']
, Tom Leinster, ''The ''n''-category Café'', March 19, 2012 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 ''x''''m'' / (''a x''2 + ''b x'' + ''c'')''n''
For
:
:
:
:
:
:
Integrands of the form ''x''''m'' (''a'' + ''b x''''n'')''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 ''x''''m'' (''A'' + ''B x''''n'') (''a'' + ''b x''''n'')''p'' (''c'' + ''d x''''n'')''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 x''2)''p'' when ''b''2 − 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 x''2)''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 ''x''''m'' (''a'' + ''b x''''n'' + ''c x''2''n'')''p'' when ''b''2 − 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 ''x''''m'' (''A'' + ''B x''''n'') (''a'' + ''b x''''n'' + ''c x''2''n'')''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
{{Lists of integrals
Rational functions
In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rati ...