- Source: List of integrals of rational functions
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:
which can then be integrated term by term.
For other types of functions, see lists of integrals.
Miscellaneous integrands
Integrands of the form xm(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 with a locally constant function. However, it is conventional to omit this from the notation. For example,
∫
1
a
x
+
b
d
x
=
{
1
a
ln
(
−
(
a
x
+
b
)
)
+
C
−
a
x
+
b
<
0
1
a
ln
(
a
x
+
b
)
+
C
+
a
x
+
b
>
0
{\displaystyle \int {\frac {1}{ax+b}}\,dx={\begin{cases}{\dfrac {1}{a}}\ln(-(ax+b))+C^{-}&ax+b<0\\{\dfrac {1}{a}}\ln(ax+b)+C^{+}&ax+b>0\end{cases}}}
is usually abbreviated as
∫
1
a
x
+
b
d
x
=
1
a
ln
|
a
x
+
b
|
+
C
,
{\displaystyle \int {\frac {1}{ax+b}}\,dx={\frac {1}{a}}\ln \left|ax+b\right|+C,}
where C is to be understood as notation for a locally constant function of x. This convention will be adhered to in the following.
Integrands of the form xm / (a x2 + b x + c)n
For
a
≠
0
:
{\displaystyle a\neq 0:}
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
(
a
+
b
x
)
m
(
c
+
d
x
)
n
(
e
+
f
x
)
p
{\displaystyle (a+b\,x)^{m}(c+d\,x)^{n}(e+f\,x)^{p}}
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
(
a
+
b
x
n
)
p
(
c
+
d
x
n
)
q
{\displaystyle \left(a+b\,x^{n}\right)^{p}\left(c+d\,x^{n}\right)^{q}}
and
x
m
(
a
+
b
x
n
)
p
(
c
+
d
x
n
)
q
{\displaystyle x^{m}\left(a+b\,x^{n}\right)^{p}\left(c+d\,x^{n}\right)^{q}}
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
(
a
+
b
x
+
c
x
2
)
p
{\displaystyle \left(a+b\,x+c\,x^{2}\right)^{p}}
when
b
2
−
4
a
c
=
0
{\displaystyle b^{2}-4\,a\,c=0}
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
(
a
+
b
x
+
c
x
2
)
p
{\displaystyle \left(a+b\,x+c\,x^{2}\right)^{p}}
and
(
d
+
e
x
)
m
(
a
+
b
x
+
c
x
2
)
p
{\displaystyle (d+e\,x)^{m}\left(a+b\,x+c\,x^{2}\right)^{p}}
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
(
a
+
b
x
n
+
c
x
2
n
)
p
{\displaystyle \left(a+b\,x^{n}+c\,x^{2n}\right)^{p}}
when
b
2
−
4
a
c
=
0
{\displaystyle b^{2}-4\,a\,c=0}
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
(
a
+
b
x
n
+
c
x
2
n
)
p
{\displaystyle \left(a+b\,x^{n}+c\,x^{2n}\right)^{p}}
and
x
m
(
a
+
b
x
n
+
c
x
2
n
)
p
{\displaystyle x^{m}\left(a+b\,x^{n}+c\,x^{2n}\right)^{p}}
by setting m and/or B to 0.
References
Kata Kunci Pencarian:
- List of integrals of rational functions
- Lists of integrals
- List of calculus topics
- List of integrals of trigonometric functions
- List of integration and measure theory topics
- List of mathematical functions
- Lists of mathematics topics
- Integral
- Polylogarithm
- Fubini's theorem
