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, so tables of known integrals are often useful. This page lists some of the most common antiderivatives .
Historical development of integrals
A compilation of a list of integrals (Integraltafeln) and techniques of integral calculus was published by the German mathematician Meier Hirsch [de ] (also spelled Meyer Hirsch) in 1810.[ 1] David Bierens de Haan for his Tables d'intégrales définies Supplément aux tables d'intégrales définies Nouvelles tables d'intégrales définies
These tables, which contain mainly integrals of elementary functions, remained in use until the middle of the 20th century. They were then replaced by the much more extensive tables of Gradshteyn and Ryzhik . In Gradshteyn and Ryzhik, integrals originating from the book by Bierens de Haan are denoted by BI.
Not all closed-form expressions have closed-form antiderivatives; this study forms the subject of differential Galois theory , which was initially developed by Joseph Liouville in the 1830s and 1840s, leading to Liouville's theorem which classifies which expressions have closed-form antiderivatives. A simple example of a function without a closed-form antiderivative is e −x 2 error function .
Since 1968 there is the Risch algorithm for determining indefinite integrals that can be expressed in term of elementary functions , typically using a computer algebra system . Integrals that cannot be expressed using elementary functions can be manipulated symbolically using general functions such as the Meijer G-function .
Lists of integrals
More detail may be found on the following pages for the lists of integrals :
Gradshteyn , Ryzhik , Geronimus , Tseytlin , Jeffrey, Zwillinger, and Moll 's (GR) Table of Integrals, Series, and Products Integrals and Series by Prudnikov , Brychkov , and Marichev (with volumes 1–3 listing integrals and series of elementary and special functions , volume 4–5 are tables of Laplace transforms ). More compact collections can be found in e.g. Brychkov, Marichev, Prudnikov's Tables of Indefinite Integrals , or as chapters in Zwillinger's CRC Standard Mathematical Tables and Formulae or Bronshtein and Semendyayev 's Guide Book to Mathematics Handbook of Mathematics Users' Guide to Mathematics
Other useful resources include Abramowitz and Stegun and the Bateman Manuscript Project . Both works contain many identities concerning specific integrals, which are organized with the most relevant topic instead of being collected into a separate table. Two volumes of the Bateman Manuscript are specific to integral transforms.
There are several web sites which have tables of integrals and integrals on demand. Wolfram Alpha can show results, and for some simpler expressions, also the intermediate steps of the integration. Wolfram Research also operates another online service, the Mathematica Online Integrator.
Integrals of simple functions
C is used for an arbitrary constant of integration that can only be determined if something about the value of the integral at some point is known. Thus, each function has an infinite number of antiderivatives .
These formulas only state in another form the assertions in the table of derivatives .
Integrals with a singularity
When there is a singularity in the function being integrated such that the antiderivative becomes undefined or at some point (the singularity), then C does not need to be the same on both sides of the singularity. The forms below normally assume the Cauchy principal value around a singularity in the value of C but this is in general, not necessary. For instance in
∫ ∫ -->
1
x
d
x
=
ln
-->
|
x
|
+
C
{\displaystyle \int {1 \over x}\,dx=\ln \left|x\right|+C}
antiderivative becomes infinite there. If the integral above were to be used to compute a definite integral between −1 and 1, one would get the wrong answer 0. This however is the Cauchy principal value of the integral around the singularity. If the integration is done in the complex plane the result depends on the path around the origin, in this case the singularity contributes −i π when using a path above the origin and i π for a path below the origin. A function on the real line could use a completely different value of C on either side of the origin as in:[ 2]
∫ ∫ -->
1
x
d
x
=
ln
-->
|
x
|
+
{
A
if
x
>
0
;
B
if
x
<
0.
{\displaystyle \int {1 \over x}\,dx=\ln |x|+{\begin{cases}A&{\text{if }}x>0;\\B&{\text{if }}x<0.\end{cases}}}
Rational functions
∫ ∫ -->
a
d
x
=
a
x
+
C
{\displaystyle \int a\,dx=ax+C}
The following function has a non-integrable singularity at 0 for n ≤ −1
∫ ∫ -->
x
n
d
x
=
x
n
+
1
n
+
1
+
C
(for
n
≠ ≠ -->
− − -->
1
)
{\displaystyle \int x^{n}\,dx={\frac {x^{n+1}}{n+1}}+C\qquad {\text{(for }}n\neq -1{\text{)}}}
Cavalieri's quadrature formula )
∫ ∫ -->
(
a
x
+
b
)
n
d
x
=
(
a
x
+
b
)
n
+
1
a
(
n
+
1
)
+
C
(for
n
≠ ≠ -->
− − -->
1
)
{\displaystyle \int (ax+b)^{n}\,dx={\frac {(ax+b)^{n+1}}{a(n+1)}}+C\qquad {\text{(for }}n\neq -1{\text{)}}}
∫ ∫ -->
1
x
d
x
=
ln
-->
|
x
|
+
C
{\displaystyle \int {1 \over x}\,dx=\ln \left|x\right|+C}
More generally,[ 3]
∫ ∫ -->
1
x
d
x
=
{
ln
-->
|
x
|
+
C
− − -->
x
<
0
ln
-->
|
x
|
+
C
+
x
>
0
{\displaystyle \int {1 \over x}\,dx={\begin{cases}\ln \left|x\right|+C^{-}&x<0\\\ln \left|x\right|+C^{+}&x>0\end{cases}}}
∫ ∫ -->
c
a
x
+
b
d
x
=
c
a
ln
-->
|
a
x
+
b
|
+
C
{\displaystyle \int {\frac {c}{ax+b}}\,dx={\frac {c}{a}}\ln \left|ax+b\right|+C}
Exponential functions
Logarithms
∫ ∫ -->
ln
-->
x
d
x
=
x
ln
-->
x
− − -->
x
+
C
=
x
(
ln
-->
x
− − -->
1
)
+
C
{\displaystyle \int \ln x\,dx=x\ln x-x+C=x(\ln x-1)+C}
∫ ∫ -->
log
a
-->
x
d
x
=
x
log
a
-->
x
− − -->
x
ln
-->
a
+
C
=
x
ln
-->
a
(
ln
-->
x
− − -->
1
)
+
C
{\displaystyle \int \log _{a}x\,dx=x\log _{a}x-{\frac {x}{\ln a}}+C={\frac {x}{\ln a}}(\ln x-1)+C}
Trigonometric functions
∫ ∫ -->
sin
-->
x
d
x
=
− − -->
cos
-->
x
+
C
{\displaystyle \int \sin {x}\,dx=-\cos {x}+C}
∫ ∫ -->
cos
-->
x
d
x
=
sin
-->
x
+
C
{\displaystyle \int \cos {x}\,dx=\sin {x}+C}
∫ ∫ -->
tan
-->
x
d
x
=
ln
-->
|
sec
-->
x
|
+
C
=
− − -->
ln
-->
|
cos
-->
x
|
+
C
{\displaystyle \int \tan {x}\,dx=\ln {\left|\sec {x}\right|}+C=-\ln {\left|\cos {x}\right|}+C}
∫ ∫ -->
cot
-->
x
d
x
=
− − -->
ln
-->
|
csc
-->
x
|
+
C
=
ln
-->
|
sin
-->
x
|
+
C
{\displaystyle \int \cot {x}\,dx=-\ln {\left|\csc {x}\right|}+C=\ln {\left|\sin {x}\right|}+C}
∫ ∫ -->
sec
-->
x
d
x
=
ln
-->
|
sec
-->
x
+
tan
-->
x
|
+
C
=
ln
-->
|
tan
-->
(
x
2
+
π π -->
4
)
|
+
C
{\displaystyle \int \sec {x}\,dx=\ln {\left|\sec {x}+\tan {x}\right|}+C=\ln \left|\tan \left({\dfrac {x}{2}}+{\dfrac {\pi }{4}}\right)\right|+C}
∫ ∫ -->
csc
-->
x
d
x
=
− − -->
ln
-->
|
csc
-->
x
+
cot
-->
x
|
+
C
=
ln
-->
|
csc
-->
x
− − -->
cot
-->
x
|
+
C
=
ln
-->
|
tan
-->
x
2
|
+
C
{\displaystyle \int \csc {x}\,dx=-\ln {\left|\csc {x}+\cot {x}\right|}+C=\ln {\left|\csc {x}-\cot {x}\right|}+C=\ln {\left|\tan {\frac {x}{2}}\right|}+C}
∫ ∫ -->
sec
2
-->
x
d
x
=
tan
-->
x
+
C
{\displaystyle \int \sec ^{2}x\,dx=\tan x+C}
∫ ∫ -->
csc
2
-->
x
d
x
=
− − -->
cot
-->
x
+
C
{\displaystyle \int \csc ^{2}x\,dx=-\cot x+C}
∫ ∫ -->
sec
-->
x
tan
-->
x
d
x
=
sec
-->
x
+
C
{\displaystyle \int \sec {x}\,\tan {x}\,dx=\sec {x}+C}
∫ ∫ -->
csc
-->
x
cot
-->
x
d
x
=
− − -->
csc
-->
x
+
C
{\displaystyle \int \csc {x}\,\cot {x}\,dx=-\csc {x}+C}
∫ ∫ -->
sin
2
-->
x
d
x
=
1
2
(
x
− − -->
sin
-->
2
x
2
)
+
C
=
1
2
(
x
− − -->
sin
-->
x
cos
-->
x
)
+
C
{\displaystyle \int \sin ^{2}x\,dx={\frac {1}{2}}\left(x-{\frac {\sin 2x}{2}}\right)+C={\frac {1}{2}}(x-\sin x\cos x)+C}
∫ ∫ -->
cos
2
-->
x
d
x
=
1
2
(
x
+
sin
-->
2
x
2
)
+
C
=
1
2
(
x
+
sin
-->
x
cos
-->
x
)
+
C
{\displaystyle \int \cos ^{2}x\,dx={\frac {1}{2}}\left(x+{\frac {\sin 2x}{2}}\right)+C={\frac {1}{2}}(x+\sin x\cos x)+C}
∫ ∫ -->
tan
2
-->
x
d
x
=
tan
-->
x
− − -->
x
+
C
{\displaystyle \int \tan ^{2}x\,dx=\tan x-x+C}
∫ ∫ -->
cot
2
-->
x
d
x
=
− − -->
cot
-->
x
− − -->
x
+
C
{\displaystyle \int \cot ^{2}x\,dx=-\cot x-x+C}
∫ ∫ -->
sec
3
-->
x
d
x
=
1
2
(
sec
-->
x
tan
-->
x
+
ln
-->
|
sec
-->
x
+
tan
-->
x
|
)
+
C
{\displaystyle \int \sec ^{3}x\,dx={\frac {1}{2}}(\sec x\tan x+\ln |\sec x+\tan x|)+C}
∫ ∫ -->
csc
3
-->
x
d
x
=
1
2
(
− − -->
csc
-->
x
cot
-->
x
+
ln
-->
|
csc
-->
x
− − -->
cot
-->
x
|
)
+
C
=
1
2
(
ln
-->
|
tan
-->
x
2
|
− − -->
csc
-->
x
cot
-->
x
)
+
C
{\displaystyle \int \csc ^{3}x\,dx={\frac {1}{2}}(-\csc x\cot x+\ln |\csc x-\cot x|)+C={\frac {1}{2}}\left(\ln \left|\tan {\frac {x}{2}}\right|-\csc x\cot x\right)+C}
∫ ∫ -->
sin
n
-->
x
d
x
=
− − -->
sin
n
− − -->
1
-->
x
cos
-->
x
n
+
n
− − -->
1
n
∫ ∫ -->
sin
n
− − -->
2
-->
x
d
x
{\displaystyle \int \sin ^{n}x\,dx=-{\frac {\sin ^{n-1}{x}\cos {x}}{n}}+{\frac {n-1}{n}}\int \sin ^{n-2}{x}\,dx}
∫ ∫ -->
cos
n
-->
x
d
x
=
cos
n
− − -->
1
-->
x
sin
-->
x
n
+
n
− − -->
1
n
∫ ∫ -->
cos
n
− − -->
2
-->
x
d
x
{\displaystyle \int \cos ^{n}x\,dx={\frac {\cos ^{n-1}{x}\sin {x}}{n}}+{\frac {n-1}{n}}\int \cos ^{n-2}{x}\,dx}
Inverse trigonometric functions
∫ ∫ -->
arcsin
-->
x
d
x
=
x
arcsin
-->
x
+
1
− − -->
x
2
+
C
,
for
|
x
|
≤ ≤ -->
1
{\displaystyle \int \arcsin {x}\,dx=x\arcsin {x}+{\sqrt {1-x^{2}}}+C,{\text{ for }}\vert x\vert \leq 1}
∫ ∫ -->
arccos
-->
x
d
x
=
x
arccos
-->
x
− − -->
1
− − -->
x
2
+
C
,
for
|
x
|
≤ ≤ -->
1
{\displaystyle \int \arccos {x}\,dx=x\arccos {x}-{\sqrt {1-x^{2}}}+C,{\text{ for }}\vert x\vert \leq 1}
∫ ∫ -->
arctan
-->
x
d
x
=
x
arctan
-->
x
− − -->
1
2
ln
-->
|
1
+
x
2
|
+
C
,
for all real
x
{\displaystyle \int \arctan {x}\,dx=x\arctan {x}-{\frac {1}{2}}\ln {\vert 1+x^{2}\vert }+C,{\text{ for all real }}x}
∫ ∫ -->
arccot
-->
x
d
x
=
x
arccot
-->
x
+
1
2
ln
-->
|
1
+
x
2
|
+
C
,
for all real
x
{\displaystyle \int \operatorname {arccot} {x}\,dx=x\operatorname {arccot} {x}+{\frac {1}{2}}\ln {\vert 1+x^{2}\vert }+C,{\text{ for all real }}x}
∫ ∫ -->
arcsec
-->
x
d
x
=
x
arcsec
-->
x
− − -->
ln
-->
|
x
(
1
+
1
− − -->
x
− − -->
2
)
|
+
C
,
for
|
x
|
≥ ≥ -->
1
{\displaystyle \int \operatorname {arcsec} {x}\,dx=x\operatorname {arcsec} {x}-\ln \left\vert x\,\left(1+{\sqrt {1-x^{-2}}}\,\right)\right\vert +C,{\text{ for }}\vert x\vert \geq 1}
∫ ∫ -->
arccsc
-->
x
d
x
=
x
arccsc
-->
x
+
ln
-->
|
x
(
1
+
1
− − -->
x
− − -->
2
)
|
+
C
,
for
|
x
|
≥ ≥ -->
1
{\displaystyle \int \operatorname {arccsc} {x}\,dx=x\operatorname {arccsc} {x}+\ln \left\vert x\,\left(1+{\sqrt {1-x^{-2}}}\,\right)\right\vert +C,{\text{ for }}\vert x\vert \geq 1}
Hyperbolic functions
∫ ∫ -->
sinh
-->
x
d
x
=
cosh
-->
x
+
C
{\displaystyle \int \sinh x\,dx=\cosh x+C}
∫ ∫ -->
cosh
-->
x
d
x
=
sinh
-->
x
+
C
{\displaystyle \int \cosh x\,dx=\sinh x+C}
∫ ∫ -->
tanh
-->
x
d
x
=
ln
(
cosh
-->
x
)
+
C
{\displaystyle \int \tanh x\,dx=\ln \,(\cosh x)+C}
∫ ∫ -->
coth
-->
x
d
x
=
ln
-->
|
sinh
-->
x
|
+
C
,
for
x
≠ ≠ -->
0
{\displaystyle \int \coth x\,dx=\ln |\sinh x|+C,{\text{ for }}x\neq 0}
∫ ∫ -->
sech
x
d
x
=
arctan
(
sinh
-->
x
)
+
C
{\displaystyle \int \operatorname {sech} \,x\,dx=\arctan \,(\sinh x)+C}
∫ ∫ -->
csch
x
d
x
=
ln
-->
|
coth
-->
x
− − -->
csch
-->
x
|
+
C
=
ln
-->
|
tanh
-->
x
2
|
+
C
,
for
x
≠ ≠ -->
0
{\displaystyle \int \operatorname {csch} \,x\,dx=\ln |\operatorname {coth} x-\operatorname {csch} x|+C=\ln \left|\tanh {x \over 2}\right|+C,{\text{ for }}x\neq 0}
∫ ∫ -->
sech
2
-->
x
d
x
=
tanh
-->
x
+
C
{\displaystyle \int \operatorname {sech} ^{2}x\,dx=\tanh x+C}
∫ ∫ -->
csch
2
-->
x
d
x
=
− − -->
coth
-->
x
+
C
{\displaystyle \int \operatorname {csch} ^{2}x\,dx=-\operatorname {coth} x+C}
∫ ∫ -->
sech
-->
x
tanh
-->
x
d
x
=
− − -->
sech
-->
x
+
C
{\displaystyle \int \operatorname {sech} {x}\,\operatorname {tanh} {x}\,dx=-\operatorname {sech} {x}+C}
∫ ∫ -->
csch
-->
x
coth
-->
x
d
x
=
− − -->
csch
-->
x
+
C
{\displaystyle \int \operatorname {csch} {x}\,\operatorname {coth} {x}\,dx=-\operatorname {csch} {x}+C}
Inverse hyperbolic functions
∫ ∫ -->
arcsinh
x
d
x
=
x
arcsinh
x
− − -->
x
2
+
1
+
C
,
for all real
x
{\displaystyle \int \operatorname {arcsinh} \,x\,dx=x\,\operatorname {arcsinh} \,x-{\sqrt {x^{2}+1}}+C,{\text{ for all real }}x}
∫ ∫ -->
arccosh
x
d
x
=
x
arccosh
x
− − -->
x
2
− − -->
1
+
C
,
for
x
≥ ≥ -->
1
{\displaystyle \int \operatorname {arccosh} \,x\,dx=x\,\operatorname {arccosh} \,x-{\sqrt {x^{2}-1}}+C,{\text{ for }}x\geq 1}
∫ ∫ -->
arctanh
x
d
x
=
x
arctanh
x
+
ln
-->
(
1
− − -->
x
2
)
2
+
C
,
for
|
x
|
<
1
{\displaystyle \int \operatorname {arctanh} \,x\,dx=x\,\operatorname {arctanh} \,x+{\frac {\ln \left(\,1-x^{2}\right)}{2}}+C,{\text{ for }}\vert x\vert <1}
∫ ∫ -->
arccoth
x
d
x
=
x
arccoth
x
+
ln
-->
(
x
2
− − -->
1
)
2
+
C
,
for
|
x
|
>
1
{\displaystyle \int \operatorname {arccoth} \,x\,dx=x\,\operatorname {arccoth} \,x+{\frac {\ln \left(x^{2}-1\right)}{2}}+C,{\text{ for }}\vert x\vert >1}
∫ ∫ -->
arcsech
x
d
x
=
x
arcsech
x
+
arcsin
-->
x
+
C
,
for
0
<
x
≤ ≤ -->
1
{\displaystyle \int \operatorname {arcsech} \,x\,dx=x\,\operatorname {arcsech} \,x+\arcsin x+C,{\text{ for }}0<x\leq 1}
∫ ∫ -->
arccsch
x
d
x
=
x
arccsch
x
+
|
arcsinh
x
|
+
C
,
for
x
≠ ≠ -->
0
{\displaystyle \int \operatorname {arccsch} \,x\,dx=x\,\operatorname {arccsch} \,x+\vert \operatorname {arcsinh} \,x\vert +C,{\text{ for }}x\neq 0}
Products of functions proportional to their second derivatives
∫ ∫ -->
cos
-->
a
x
e
b
x
d
x
=
e
b
x
a
2
+
b
2
(
a
sin
-->
a
x
+
b
cos
-->
a
x
)
+
C
{\displaystyle \int \cos ax\,e^{bx}\,dx={\frac {e^{bx}}{a^{2}+b^{2}}}\left(a\sin ax+b\cos ax\right)+C}
∫ ∫ -->
sin
-->
a
x
e
b
x
d
x
=
e
b
x
a
2
+
b
2
(
b
sin
-->
a
x
− − -->
a
cos
-->
a
x
)
+
C
{\displaystyle \int \sin ax\,e^{bx}\,dx={\frac {e^{bx}}{a^{2}+b^{2}}}\left(b\sin ax-a\cos ax\right)+C}
∫ ∫ -->
cos
-->
a
x
cosh
-->
b
x
d
x
=
1
a
2
+
b
2
(
a
sin
-->
a
x
cosh
-->
b
x
+
b
cos
-->
a
x
sinh
-->
b
x
)
+
C
{\displaystyle \int \cos ax\,\cosh bx\,dx={\frac {1}{a^{2}+b^{2}}}\left(a\sin ax\,\cosh bx+b\cos ax\,\sinh bx\right)+C}
∫ ∫ -->
sin
-->
a
x
cosh
-->
b
x
d
x
=
1
a
2
+
b
2
(
b
sin
-->
a
x
sinh
-->
b
x
− − -->
a
cos
-->
a
x
cosh
-->
b
x
)
+
C
{\displaystyle \int \sin ax\,\cosh bx\,dx={\frac {1}{a^{2}+b^{2}}}\left(b\sin ax\,\sinh bx-a\cos ax\,\cosh bx\right)+C}
Absolute-value functions
Let f continuous function , that has at most one zero . If f g f f g f
∫ ∫ -->
|
f
(
x
)
|
d
x
=
sgn
-->
(
f
(
x
)
)
g
(
x
)
+
C
,
{\displaystyle \int \left|f(x)\right|\,dx=\operatorname {sgn}(f(x))g(x)+C,}
sgn(x ) is the sign function , which takes the values −1, 0, 1 when x
This can be proved by computing the derivative of the right-hand side of the formula, taking into account that the condition on g
This gives the following formulas (where a ≠ 0f C must be replaced by a piecewise constant function):
∫ ∫ -->
|
(
a
x
+
b
)
n
|
d
x
=
sgn
-->
(
a
x
+
b
)
(
a
x
+
b
)
n
+
1
a
(
n
+
1
)
+
C
{\displaystyle \int \left|(ax+b)^{n}\right|\,dx=\operatorname {sgn}(ax+b){(ax+b)^{n+1} \over a(n+1)}+C}
when n
n
≠ ≠ -->
− − -->
1
{\displaystyle n\neq -1}
∫ ∫ -->
|
tan
-->
a
x
|
d
x
=
− − -->
1
a
sgn
-->
(
tan
-->
a
x
)
ln
-->
(
|
cos
-->
a
x
|
)
+
C
{\displaystyle \int \left|\tan {ax}\right|\,dx=-{\frac {1}{a}}\operatorname {sgn}(\tan {ax})\ln(\left|\cos {ax}\right|)+C}
when
a
x
∈ ∈ -->
(
n
π π -->
− − -->
π π -->
2
,
n
π π -->
+
π π -->
2
)
{\textstyle ax\in \left(n\pi -{\frac {\pi }{2}},n\pi +{\frac {\pi }{2}}\right)}
n
∫ ∫ -->
|
csc
-->
a
x
|
d
x
=
− − -->
1
a
sgn
-->
(
csc
-->
a
x
)
ln
-->
(
|
csc
-->
a
x
+
cot
-->
a
x
|
)
+
C
{\displaystyle \int \left|\csc {ax}\right|\,dx=-{\frac {1}{a}}\operatorname {sgn}(\csc {ax})\ln(\left|\csc {ax}+\cot {ax}\right|)+C}
when
a
x
∈ ∈ -->
(
n
π π -->
,
n
π π -->
+
π π -->
)
{\displaystyle ax\in \left(n\pi ,n\pi +\pi \right)}
n
∫ ∫ -->
|
sec
-->
a
x
|
d
x
=
1
a
sgn
-->
(
sec
-->
a
x
)
ln
-->
(
|
sec
-->
a
x
+
tan
-->
a
x
|
)
+
C
{\displaystyle \int \left|\sec {ax}\right|\,dx={\frac {1}{a}}\operatorname {sgn}(\sec {ax})\ln(\left|\sec {ax}+\tan {ax}\right|)+C}
when
a
x
∈ ∈ -->
(
n
π π -->
− − -->
π π -->
2
,
n
π π -->
+
π π -->
2
)
{\textstyle ax\in \left(n\pi -{\frac {\pi }{2}},n\pi +{\frac {\pi }{2}}\right)}
n
∫ ∫ -->
|
cot
-->
a
x
|
d
x
=
1
a
sgn
-->
(
cot
-->
a
x
)
ln
-->
(
|
sin
-->
a
x
|
)
+
C
{\displaystyle \int \left|\cot {ax}\right|\,dx={\frac {1}{a}}\operatorname {sgn}(\cot {ax})\ln(\left|\sin {ax}\right|)+C}
when
a
x
∈ ∈ -->
(
n
π π -->
,
n
π π -->
+
π π -->
)
{\displaystyle ax\in \left(n\pi ,n\pi +\pi \right)}
n If the function f f sgn(f (x )) ∫ f (x ) dx is an antiderivative of f interval on which f f (x ) = 0step function . If we also use the fact that the absolute values of sine and cosine are periodic with period π , then we get:
∫ ∫ -->
|
sin
-->
a
x
|
d
x
=
2
a
⌊
a
x
π π -->
⌋
− − -->
1
a
cos
-->
(
a
x
− − -->
⌊
a
x
π π -->
⌋
π π -->
)
+
C
{\displaystyle \int \left|\sin {ax}\right|\,dx={2 \over a}\left\lfloor {\frac {ax}{\pi }}\right\rfloor -{1 \over a}\cos {\left(ax-\left\lfloor {\frac {ax}{\pi }}\right\rfloor \pi \right)}+C}
[citation needed
∫ ∫ -->
|
cos
-->
a
x
|
d
x
=
2
a
⌊
a
x
π π -->
+
1
2
⌋
+
1
a
sin
-->
(
a
x
− − -->
⌊
a
x
π π -->
+
1
2
⌋
π π -->
)
+
C
{\displaystyle \int \left|\cos {ax}\right|\,dx={2 \over a}\left\lfloor {\frac {ax}{\pi }}+{\frac {1}{2}}\right\rfloor +{1 \over a}\sin {\left(ax-\left\lfloor {\frac {ax}{\pi }}+{\frac {1}{2}}\right\rfloor \pi \right)}+C}
[citation needed
Special functions
Ci , Si : Trigonometric integrals , Ei : Exponential integral , li : Logarithmic integral function , erf : Error function
∫ ∫ -->
Ci
-->
(
x
)
d
x
=
x
Ci
-->
(
x
)
− − -->
sin
-->
x
{\displaystyle \int \operatorname {Ci} (x)\,dx=x\operatorname {Ci} (x)-\sin x}
∫ ∫ -->
Si
-->
(
x
)
d
x
=
x
Si
-->
(
x
)
+
cos
-->
x
{\displaystyle \int \operatorname {Si} (x)\,dx=x\operatorname {Si} (x)+\cos x}
∫ ∫ -->
Ei
-->
(
x
)
d
x
=
x
Ei
-->
(
x
)
− − -->
e
x
{\displaystyle \int \operatorname {Ei} (x)\,dx=x\operatorname {Ei} (x)-e^{x}}
∫ ∫ -->
li
-->
(
x
)
d
x
=
x
li
-->
(
x
)
− − -->
Ei
-->
(
2
ln
-->
x
)
{\displaystyle \int \operatorname {li} (x)\,dx=x\operatorname {li} (x)-\operatorname {Ei} (2\ln x)}
∫ ∫ -->
li
-->
(
x
)
x
d
x
=
ln
-->
x
li
-->
(
x
)
− − -->
x
{\displaystyle \int {\frac {\operatorname {li} (x)}{x}}\,dx=\ln x\,\operatorname {li} (x)-x}
∫ ∫ -->
erf
-->
(
x
)
d
x
=
e
− − -->
x
2
π π -->
+
x
erf
-->
(
x
)
{\displaystyle \int \operatorname {erf} (x)\,dx={\frac {e^{-x^{2}}}{\sqrt {\pi }}}+x\operatorname {erf} (x)}
There are some functions whose antiderivatives cannot be expressed in closed form . However, the values of the definite integrals of some of these functions over some common intervals can be calculated. A few useful integrals are given below.
∫ ∫ -->
0
∞ ∞ -->
x
e
− − -->
x
d
x
=
1
2
π π -->
{\displaystyle \int _{0}^{\infty }{\sqrt {x}}\,e^{-x}\,dx={\frac {1}{2}}{\sqrt {\pi }}}
Gamma function )
∫ ∫ -->
0
∞ ∞ -->
e
− − -->
a
x
2
d
x
=
1
2
π π -->
a
{\displaystyle \int _{0}^{\infty }e^{-ax^{2}}\,dx={\frac {1}{2}}{\sqrt {\frac {\pi }{a}}}}
a > 0Gaussian integral )
∫ ∫ -->
0
∞ ∞ -->
x
2
e
− − -->
a
x
2
d
x
=
1
4
π π -->
a
3
{\displaystyle \int _{0}^{\infty }{x^{2}e^{-ax^{2}}\,dx}={\frac {1}{4}}{\sqrt {\frac {\pi }{a^{3}}}}}
a > 0
∫ ∫ -->
0
∞ ∞ -->
x
2
n
e
− − -->
a
x
2
d
x
=
2
n
− − -->
1
2
a
∫ ∫ -->
0
∞ ∞ -->
x
2
(
n
− − -->
1
)
e
− − -->
a
x
2
d
x
=
(
2
n
− − -->
1
)
!
!
2
n
+
1
π π -->
a
2
n
+
1
=
(
2
n
)
!
n
!
2
2
n
+
1
π π -->
a
2
n
+
1
{\displaystyle \int _{0}^{\infty }x^{2n}e^{-ax^{2}}\,dx={\frac {2n-1}{2a}}\int _{0}^{\infty }x^{2(n-1)}e^{-ax^{2}}\,dx={\frac {(2n-1)!!}{2^{n+1}}}{\sqrt {\frac {\pi }{a^{2n+1}}}}={\frac {(2n)!}{n!2^{2n+1}}}{\sqrt {\frac {\pi }{a^{2n+1}}}}}
for a > 0n !! is the double factorial .
∫ ∫ -->
0
∞ ∞ -->
x
3
e
− − -->
a
x
2
d
x
=
1
2
a
2
{\displaystyle \int _{0}^{\infty }{x^{3}e^{-ax^{2}}\,dx}={\frac {1}{2a^{2}}}}
a > 0
∫ ∫ -->
0
∞ ∞ -->
x
2
n
+
1
e
− − -->
a
x
2
d
x
=
n
a
∫ ∫ -->
0
∞ ∞ -->
x
2
n
− − -->
1
e
− − -->
a
x
2
d
x
=
n
!
2
a
n
+
1
{\displaystyle \int _{0}^{\infty }x^{2n+1}e^{-ax^{2}}\,dx={\frac {n}{a}}\int _{0}^{\infty }x^{2n-1}e^{-ax^{2}}\,dx={\frac {n!}{2a^{n+1}}}}
for a > 0n = 0, 1, 2, ....
∫ ∫ -->
0
∞ ∞ -->
x
e
x
− − -->
1
d
x
=
π π -->
2
6
{\displaystyle \int _{0}^{\infty }{\frac {x}{e^{x}-1}}\,dx={\frac {\pi ^{2}}{6}}}
Bernoulli number )
∫ ∫ -->
0
∞ ∞ -->
x
2
e
x
− − -->
1
d
x
=
2
ζ ζ -->
(
3
)
≈ ≈ -->
2.40
{\displaystyle \int _{0}^{\infty }{\frac {x^{2}}{e^{x}-1}}\,dx=2\zeta (3)\approx 2.40}
∫ ∫ -->
0
∞ ∞ -->
x
3
e
x
− − -->
1
d
x
=
π π -->
4
15
{\displaystyle \int _{0}^{\infty }{\frac {x^{3}}{e^{x}-1}}\,dx={\frac {\pi ^{4}}{15}}}
∫ ∫ -->
0
∞ ∞ -->
sin
-->
x
x
d
x
=
π π -->
2
{\displaystyle \int _{0}^{\infty }{\frac {\sin {x}}{x}}\,dx={\frac {\pi }{2}}}
sinc function and the Dirichlet integral )
∫ ∫ -->
0
∞ ∞ -->
sin
2
-->
x
x
2
d
x
=
π π -->
2
{\displaystyle \int _{0}^{\infty }{\frac {\sin ^{2}{x}}{x^{2}}}\,dx={\frac {\pi }{2}}}
∫ ∫ -->
0
π π -->
2
sin
n
-->
x
d
x
=
∫ ∫ -->
0
π π -->
2
cos
n
-->
x
d
x
=
(
n
− − -->
1
)
!
!
n
!
!
× × -->
{
1
if
n
is odd
π π -->
2
if
n
is even.
{\displaystyle \int _{0}^{\frac {\pi }{2}}\sin ^{n}x\,dx=\int _{0}^{\frac {\pi }{2}}\cos ^{n}x\,dx={\frac {(n-1)!!}{n!!}}\times {\begin{cases}1&{\text{if }}n{\text{ is odd}}\\{\frac {\pi }{2}}&{\text{if }}n{\text{ is even.}}\end{cases}}}
(if n double factorial ).
∫ ∫ -->
− − -->
π π -->
π π -->
cos
-->
(
α α -->
x
)
cos
n
-->
(
β β -->
x
)
d
x
=
{
2
π π -->
2
n
(
n
m
)
|
α α -->
|
=
|
β β -->
(
2
m
− − -->
n
)
|
0
otherwise
{\displaystyle \int _{-\pi }^{\pi }\cos(\alpha x)\cos ^{n}(\beta x)dx={\begin{cases}{\frac {2\pi }{2^{n}}}{\binom {n}{m}}&|\alpha |=|\beta (2m-n)|\\0&{\text{otherwise}}\end{cases}}}
(for α , β , m , n β ≠ 0m , n ≥ 0Binomial coefficient )
∫ ∫ -->
− − -->
t
t
sin
m
-->
(
α α -->
x
)
cos
n
-->
(
β β -->
x
)
d
x
=
0
{\displaystyle \int _{-t}^{t}\sin ^{m}(\alpha x)\cos ^{n}(\beta x)dx=0}
(for α , β n m an odd, positive integer; since the integrand is odd )
∫ ∫ -->
− − -->
π π -->
π π -->
sin
-->
(
α α -->
x
)
sin
n
-->
(
β β -->
x
)
d
x
=
{
(
− − -->
1
)
(
n
+
1
2
)
(
− − -->
1
)
m
2
π π -->
2
n
(
n
m
)
n
odd
,
α α -->
=
β β -->
(
2
m
− − -->
n
)
0
otherwise
{\displaystyle \int _{-\pi }^{\pi }\sin(\alpha x)\sin ^{n}(\beta x)dx={\begin{cases}(-1)^{\left({\frac {n+1}{2}}\right)}(-1)^{m}{\frac {2\pi }{2^{n}}}{\binom {n}{m}}&n{\text{ odd}},\ \alpha =\beta (2m-n)\\0&{\text{otherwise}}\end{cases}}}
(for α , β , m , n β ≠ 0m , n ≥ 0Binomial coefficient )
∫ ∫ -->
− − -->
π π -->
π π -->
cos
-->
(
α α -->
x
)
sin
n
-->
(
β β -->
x
)
d
x
=
{
(
− − -->
1
)
(
n
2
)
(
− − -->
1
)
m
2
π π -->
2
n
(
n
m
)
n
even
,
|
α α -->
|
=
|
β β -->
(
2
m
− − -->
n
)
|
0
otherwise
{\displaystyle \int _{-\pi }^{\pi }\cos(\alpha x)\sin ^{n}(\beta x)dx={\begin{cases}(-1)^{\left({\frac {n}{2}}\right)}(-1)^{m}{\frac {2\pi }{2^{n}}}{\binom {n}{m}}&n{\text{ even}},\ |\alpha |=|\beta (2m-n)|\\0&{\text{otherwise}}\end{cases}}}
(for α , β , m , n β ≠ 0m , n ≥ 0Binomial coefficient )
∫ ∫ -->
− − -->
∞ ∞ -->
∞ ∞ -->
e
− − -->
(
a
x
2
+
b
x
+
c
)
d
x
=
π π -->
a
exp
-->
[
b
2
− − -->
4
a
c
4
a
]
{\displaystyle \int _{-\infty }^{\infty }e^{-(ax^{2}+bx+c)}\,dx={\sqrt {\frac {\pi }{a}}}\exp \left[{\frac {b^{2}-4ac}{4a}}\right]}
(where exp[u ] is the exponential function e u a > 0
∫ ∫ -->
0
∞ ∞ -->
x
z
− − -->
1
e
− − -->
x
d
x
=
Γ Γ -->
(
z
)
{\displaystyle \int _{0}^{\infty }x^{z-1}\,e^{-x}\,dx=\Gamma (z)}
(where
Γ Γ -->
(
z
)
{\displaystyle \Gamma (z)}
Gamma function )
∫ ∫ -->
0
1
(
ln
-->
1
x
)
p
d
x
=
Γ Γ -->
(
p
+
1
)
{\displaystyle \int _{0}^{1}\left(\ln {\frac {1}{x}}\right)^{p}\,dx=\Gamma (p+1)}
∫ ∫ -->
0
1
x
α α -->
− − -->
1
(
1
− − -->
x
)
β β -->
− − -->
1
d
x
=
Γ Γ -->
(
α α -->
)
Γ Γ -->
(
β β -->
)
Γ Γ -->
(
α α -->
+
β β -->
)
{\displaystyle \int _{0}^{1}x^{\alpha -1}(1-x)^{\beta -1}dx={\frac {\Gamma (\alpha )\Gamma (\beta )}{\Gamma (\alpha +\beta )}}}
(for Re(α ) > 0 and Re(β ) > 0 , see Beta function )
∫ ∫ -->
0
2
π π -->
e
x
cos
-->
θ θ -->
d
θ θ -->
=
2
π π -->
I
0
(
x
)
{\displaystyle \int _{0}^{2\pi }e^{x\cos \theta }d\theta =2\pi I_{0}(x)}
I 0 (x )Bessel function of the first kind)
∫ ∫ -->
0
2
π π -->
e
x
cos
-->
θ θ -->
+
y
sin
-->
θ θ -->
d
θ θ -->
=
2
π π -->
I
0
(
x
2
+
y
2
)
{\displaystyle \int _{0}^{2\pi }e^{x\cos \theta +y\sin \theta }d\theta =2\pi I_{0}\left({\sqrt {x^{2}+y^{2}}}\right)}
∫ ∫ -->
− − -->
∞ ∞ -->
∞ ∞ -->
(
1
+
x
2
ν ν -->
)
− − -->
ν ν -->
+
1
2
d
x
=
ν ν -->
π π -->
Γ Γ -->
(
ν ν -->
2
)
Γ Γ -->
(
ν ν -->
+
1
2
)
{\displaystyle \int _{-\infty }^{\infty }\left(1+{\frac {x^{2}}{\nu }}\right)^{-{\frac {\nu +1}{2}}}\,dx={\frac {{\sqrt {\nu \pi }}\ \Gamma \left({\frac {\nu }{2}}\right)}{\Gamma \left({\frac {\nu +1}{2}}\right)}}}
(for ν > 0probability density function of Student's t -distribution )If the function f bounded variation on the interval [a ,b ] , then the method of exhaustion provides a formula for the integral:
∫ ∫ -->
a
b
f
(
x
)
d
x
=
(
b
− − -->
a
)
∑ ∑ -->
n
=
1
∞ ∞ -->
∑ ∑ -->
m
=
1
2
n
− − -->
1
(
− − -->
1
)
m
+
1
2
− − -->
n
f
(
a
+
m
(
b
− − -->
a
)
2
− − -->
n
)
.
{\displaystyle \int _{a}^{b}{f(x)\,dx}=(b-a)\sum \limits _{n=1}^{\infty }{\sum \limits _{m=1}^{2^{n}-1}{\left({-1}\right)^{m+1}}}2^{-n}f(a+m\left({b-a}\right)2^{-n}).}
The "sophomore's dream ":
∫ ∫ -->
0
1
x
− − -->
x
d
x
=
∑ ∑ -->
n
=
1
∞ ∞ -->
n
− − -->
n
(
=
1.29128
59970
6266
… … -->
)
∫ ∫ -->
0
1
x
x
d
x
=
− − -->
∑ ∑ -->
n
=
1
∞ ∞ -->
(
− − -->
n
)
− − -->
n
(
=
0.78343
05107
1213
… … -->
)
{\displaystyle {\begin{aligned}\int _{0}^{1}x^{-x}\,dx&=\sum _{n=1}^{\infty }n^{-n}&&(=1.29128\,59970\,6266\dots )\\[6pt]\int _{0}^{1}x^{x}\,dx&=-\sum _{n=1}^{\infty }(-n)^{-n}&&(=0.78343\,05107\,1213\dots )\end{aligned}}}
Johann Bernoulli .
See also
References
Further reading
Abramowitz, Milton ; Stegun, Irene Ann , eds. (1983) [June 1964]. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables ISBN 978-0-486-61272-0 LCCN 64-60036 . MR 0167642 . LCCN 65-12253 .Bronstein, Ilja Nikolaevič; Semendjajew, Konstantin Adolfovič (1987) [1945]. Grosche, Günter; Ziegler, Viktor; Ziegler, Dorothea (eds.). Taschenbuch der Mathematik Verlag Harri Deutsch (and B. G. Teubner Verlagsgesellschaft , Leipzig). ISBN 3-87144-492-8 Gradshteyn, Izrail Solomonovich ; Ryzhik, Iosif Moiseevich ; Geronimus, Yuri Veniaminovich ; Tseytlin, Michail Yulyevich ; Jeffrey, Alan (2015) [October 2014]. Zwillinger, Daniel; Moll, Victor Hugo (eds.). Table of Integrals, Series, and Products Academic Press, Inc. ISBN 978-0-12-384933-5 LCCN 2014010276 .Prudnikov, Anatolii Platonovich (Прудников, Анатолий Платонович) ; Brychkov, Yuri A. (Брычков, Ю. А.); Marichev, Oleg Igorevich (Маричев, Олег Игоревич) (1988–1992) [1981−1986 (Russian)]. Integrals and Series . Vol. 1–5. Translated by Queen, N. M. (1 ed.). (Nauka ) Gordon & Breach Science Publishers/CRC Press . ISBN 2-88124-097-6 {{cite book }}: CS1 maint: multiple names: authors list (link )Yuri A. Brychkov (Ю. А. Брычков), Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas . Russian edition, Fiziko-Matematicheskaya Literatura, 2006. English edition, Chapman & Hall/CRC Press, 2008, ISBN 1-58488-956-X / 9781584889564.
Daniel Zwillinger. CRC Standard Mathematical Tables and Formulae , 31st edition. Chapman & Hall/CRC Press, 2002. ISBN 1-58488-291-3 . (Many earlier editions as well.)
Meyer Hirsch [de ] , Integraltafeln oder Sammlung von Integralformeln Meyer Hirsch [de ] , Integral Tables Or A Collection of Integral Formulae Integraltafeln ]David Bierens de Haan , Nouvelles Tables d'Intégrales définies (Engels, Leiden, 1862)Benjamin O. Pierce A short table of integrals - revised edition (Ginn & co., Boston, 1899)
External links
Tables of integrals
Derivations
Online service
Open source programs
Videos
![{\displaystyle \int _{-\infty }^{\infty }e^{-(ax^{2}+bx+c)}\,dx={\sqrt {\frac {\pi }{a}}}\exp \left[{\frac {b^{2}-4ac}{4a}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9aac3de5d20ec70a35c8d52bafaa6a82c148b566)
![{\displaystyle {\begin{aligned}\int _{0}^{1}x^{-x}\,dx&=\sum _{n=1}^{\infty }n^{-n}&&(=1.29128\,59970\,6266\dots )\\[6pt]\int _{0}^{1}x^{x}\,dx&=-\sum _{n=1}^{\infty }(-n)^{-n}&&(=0.78343\,05107\,1213\dots )\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3eb1c1f34b4606f5ebf9e88ecdb80a7681b8fcce)
