List of definite integrals

Short description: none

This is a dynamic list and may never be able to satisfy particular standards for completeness. You can help by expanding it with reliably sourced entries.

Part of a series of articles about

Calculus

* Fundamental theorem

* Limits of functions
* Continuity

* Mean value theorem
* Rolle's theorem

Differential

|  Definitions

* Derivative (generalizations)

* Differential
* infinitesimal
* of a function
* total

Concepts

* Differentiation notation
* Second derivative
* Third derivative
* Change of variables
* Implicit differentiation
* Related rates
* Taylor's theorem

Rules and identities

* Sum
* Product
* Chain
* Power
* Quotient
* Inverse
* General Leibniz
* Faà di Bruno's formula

Integral

* Lists of integrals

Definitions

* Antiderivative
* Integral (improper)
* Riemann integral
* Lebesgue integration
* Contour integration

Integration by

* Parts
* Discs
* Cylindrical shells
* Substitution (trigonometric)
* Partial fractions
* Order
* Reduction formulae

Series

* Geometric (arithmetico-geometric)
* Harmonic
* Alternating
* Power
* Binomial
* Taylor

Convergence tests

* Summand limit (term test)
* Ratio
* Root
* Integral
* Direct comparison

Limit comparison
* Alternating series
* Cauchy condensation
* Dirichlet
* Abel

Vector

* Gradient
* Divergence
* Curl
* Laplacian
* Directional derivative
* Identities

Theorems

* Divergence
* Gradient
* Green's
* Kelvin–Stokes
* Stokes

Multivariable

|  Formalisms

* Matrix
* Tensor
* Exterior
* Geometric

Definitions

* Partial derivative
* Multiple integral
* Line integral
* Surface integral
* Volume integral
* Jacobian
* Hessian

Specialized

* Fractional
* Malliavin
* Stochastic
* Variations

Glossary of calculus

* Glossary of calculus

* v
* t
* e

In mathematics, the definite integral

∫abf(x)dx

is the area of the region in the xy-plane bounded by the graph of f, the x-axis, and the lines x = a and x = b, such that area above the x-axis adds to the total, and that below the x-axis subtracts from the total.

The fundamental theorem of calculus establishes the relationship between indefinite and definite integrals and introduces a technique for evaluating definite integrals.

If the interval is infinite the definite integral is called an improper integral and defined by using appropriate limiting procedures. for example:

∫a∞f(x)dx=limb->∞[∫abf(x)dx]

A constant, such pi, that may be defined by the integral of an algebraic function over an algebraic domain is known as a period.

The following is a list of some of the most common or interesting definite integrals. For a list of indefinite integrals see List of indefinite integrals.

## Contents

* 1 Definite integrals involving rational or irrational expressions
* 2 Definite integrals involving trigonometric functions
* 3 Definite integrals involving exponential functions
* 4 Definite integrals involving logarithmic functions
* 5 Definite integrals involving hyperbolic functions
* 6 Frullani integrals
* 7 See also
* 8 References

## Definite integrals involving rational or irrational expressions

∫0∞dx1+xp=π/psin⁡(π/p)for ℜ(p)>1
∫0∞dxap+xp=πpap−1sin⁡(πp)for ℜ(p)>1, ℜ(a)>0
∫0∞xp−1dx1+x=πsin⁡(pπ)for 0<p<1
∫0∞xmdxxn+an=πam−n+1nsin⁡(m+1nπ)for 0<m+1<n
∫0∞xmdx1+2xcos⁡β+x2=πsin⁡(mπ)⋅sin⁡(mβ)sin⁡(β)
∫0adxa2−x2=π2
∫0aa2−x2dx=πa24
∫0axm(an−xn)pdx=am+1+npΓ(m+1n)Γ(p+1)nΓ(m+1n+p+1)
∫0∞xmdx(xn+an)r=(−1)r−1πam+1−nrΓ(m+1n)nsin⁡(m+1nπ)(r−1)!Γ(m+1n−r+1)for n(r−2)<m+1<nr

## Definite integrals involving trigonometric functions

∫0πsin⁡(mx)sin⁡(nx)dx={0if m≠nπ2if m=nfor m,n positive integers
∫0πcos⁡(mx)cos⁡(nx)dx={0if m≠nπ2if m=nfor m,n positive integers
∫0πsin⁡(mx)cos⁡(nx)dx={0if m+n even2mm2−n2if m+n oddfor m,n integers.
∫0π2sin2(x)dx=∫0π2cos2(x)dx=π4
∫0π2sin2m(x)dx=∫0π2cos2m(x)dx=1×3×5×⋯×(2m−1)2×4×6×⋯×2m⋅π2for m=1,2,3…
∫0xsin2m(t)dt=(2m−1)!!(2m)!!(x−sin⁡(x)cos⁡(x)(1+∑k=1∞sin2k(x)(2k)!!(2k+1)!!))for m=1,2,3…
∫0xcos2m(t)dt=(2m−1)!!(2m)!!(x−sin⁡(x)cos⁡(x)(1+∑k=1∞cos2k(x)(2k)!!(2k+1)!!))for m=1,2,3…
∫0π2sin2m+1(x)dx=∫0π2cos2m+1(x)dx=2×4×6×⋯×2m1×3×5×⋯×(2m+1)for m=1,2,3…
∫0π2sin2p−1(x)cos2q−1(x)dx=Γ(p)Γ(q)2Γ(p+q)=12B(p,q)
∫0∞sin⁡(px)xdx={π2if p>00if p=0−π2if p<0 (see Dirichlet integral)
∫0∞sin⁡pxcos⁡qxx dx={0 if q>p>0π2 if 0<q<pπ4 if p=q>0
∫0∞sin⁡pxsin⁡qxx2 dx={πp2 if 0<p≤qπq2 if 0<q≤p
∫0∞sin2pxx2 dx=πp2
∫0∞1−cos⁡pxx2 dx=πp2
∫0∞cos⁡px−cos⁡qxx dx=ln⁡qp
∫0∞cos⁡px−cos⁡qxx2 dx=π(q−p)2
∫0∞cos⁡mxx2+a2 dx=π2ae−ma
∫0∞xsin⁡mxx2+a2 dx=π2e−ma
∫0∞sin⁡mxx(x2+a2) dx=π2a2(1−e−ma)
∫02πdxa+bsin⁡x=2πa2−b2
∫02πdxa+bcos⁡x=2πa2−b2
∫0π2dxa+bcos⁡x=cos−1(ba)a2−b2
∫02πdx(a+bsin⁡x)2=∫02πdx(a+bcos⁡x)2=2πa(a2−b2)3/2
∫02πdx1−2acos⁡x+a2=2π1−a2for 0<a<1
∫0πxsin⁡x dx1−2acos⁡x+a2={πaln⁡|1+a|if |a|<1πaln⁡|1+1a|if |a|>1
∫0πcos⁡mx dx1−2acos⁡x+a2=πam1−a2for a2<1 , m=0,1,2,…
∫0∞sin⁡ax2 dx=∫0∞cos⁡ax2=12π2a
∫0∞sin⁡axn dx=1na1/nΓ(1n)sin⁡π2nfor n>1
∫0∞cos⁡axn dx=1na1/nΓ(1n)cos⁡π2nfor n>1
∫0∞sin⁡xx dx=∫0∞cos⁡xx dx=π2
∫0∞sin⁡xxp dx=π2Γ(p)sin⁡(pπ2)for 0<p<1
∫0∞cos⁡xxp dx=π2Γ(p)cos⁡(pπ2)for 0<p<1
∫0∞sin⁡ax2cos⁡2bx dx=12π2a(cos⁡b2a−sin⁡b2a)
∫0∞cos⁡ax2cos⁡2bx dx=12π2a(cos⁡b2a+sin⁡b2a)

## Definite integrals involving exponential functions

∫0∞xe−xdx=12π (see also Gamma function)
∫0∞e−axcos⁡bxdx=aa2+b2
∫0∞e−axsin⁡bxdx=ba2+b2
∫0∞e−axsin⁡bxxdx=tan−1ba
∫0∞e−ax−e−bxxdx=ln⁡ba
∫0∞e−ax−cos⁡(bx)xdx=ln⁡ba
∫0∞e−ax2dx=12πafor a>0 (the Gaussian integral)
∫0∞e−ax2cos⁡bxdx=12πae(−b24a)
∫0∞e−(ax2+bx+c)dx=12πae(b2−4ac4a)⋅erfc⁡b2a, where erfc⁡(p)=2π∫p∞e−x2dx
∫−∞∞e−(ax2+bx+c) dx=πae(b2−4ac4a)
∫0∞xne−ax dx=Γ(n+1)an+1
∫0∞x2e−ax2dx=14πa3for a>0
∫0∞x2ne−ax2dx=2n−12a∫0∞x2(n−1)e−ax2dx=(2n−1)!!2n+1πa2n+1=(2n)!n!22n+1πa2n+1for a>0 , n=1,2,3… (where !! is the double factorial)
∫0∞x3e−ax2dx=12a2for a>0
∫0∞x2n+1e−ax2dx=na∫0∞x2n−1e−ax2dx=n!2an+1for a>0 , n=0,1,2…
∫0∞xme−ax2 dx=Γ(m+12)2a(m+12)
∫0∞xne−axbdx=1b a−n+1bΓ(n+1b)
∫0∞e(−ax2−bx2) dx=12πae−2ab
∫0∞xex−1 dx=ζ(2)=π26
∫0∞xn−1ex−1 dx=Γ(n)ζ(n)
∫0∞xex+1 dx=112−122+132−142+…=π212
∫0∞xnex+1 dx=n!⋅(2n−12n)ζ(n+1)
∫0∞sin⁡mxe2πx−1 dx=14coth⁡m2−12m
∫0∞(11+x−e−x) dxx=γ (where γ is Euler–Mascheroni constant)
∫0∞e−x2−e−xx dx=γ2
∫0∞(1ex−1−e−xx) dx=γ
∫0∞e−ax−e−bxxsec⁡px dx=12ln⁡b2+p2a2+p2
∫0∞e−ax−e−bxxcsc⁡px dx=tan−1bp−tan−1ap
∫0∞e−ax(1−cos⁡x)x2 dx=cot−1a−a2ln⁡|a2+1a2|
∫−∞∞e−x2dx=π
∫−∞∞x2(n+1)e−12x2dx=(2n+1)!2nn!2πfor n=0,1,2,…

## Definite integrals involving logarithmic functions

∫01xm(ln⁡x)ndx=(−1)nn!(m+1)n+1for m>−1,n=0,1,2,…
∫1∞xm(ln⁡x)ndx=(−1)n+1n!(m+1)n+1for m<−1,n=0,1,2,…
∫01ln⁡x1+xdx=−π212
∫01ln⁡x1−xdx=−π26
∫01ln⁡(1+x)xdx=π212
∫01ln⁡(1−x)xdx=−π26
∫0∞ln⁡(a2+x2)b2+x2 dx=πbln⁡(a+b)for a,b>0
∫0∞ln⁡xx2+a2 dx=πln⁡a2afor a>0

## Definite integrals involving hyperbolic functions

∫0∞sin⁡axsinh⁡bx dx=π2btanh⁡aπ2b

∫0∞cos⁡axcosh⁡bx dx=π2b⋅1cosh⁡aπ2b

∫0∞xsinh⁡ax dx=π24a2

∫0∞x2n+1sinh⁡ax dx=c2n+1(πa)2(n+1),c2n+1=(−1)n2(12−∑k=0n−1(−1)k(2n+12k+1)c2k+1),c1=14

∫0∞1cosh⁡ax dx=π2a

∫0∞x2ncosh⁡ax dx=d2n(πa)2n+1,d2n=(−1)n2(14n−∑k=0n−1(−1)k(2n2k)d2k),d0=12

## Frullani integrals

∫0∞f(ax)−f(bx)x dx=(limx->0f(x)−limx->∞f(x))ln⁡(ba) holds if the integral exists and f′(x) is continuous.

## See also

* List of integrals
* Indefinite sum
* Gamma function
* List of limits

## References

* "Derivation of Logarithmic and Logarithmic Hyperbolic Tangent Integrals Expressed in Terms of Special Functions". Mathematics 8 (687): 687. 2020. doi:10.3390/math8050687.
* "A Definite Integral Involving the Logarithmic Function in Terms of the Lerch Function". Mathematics 7 (1148): 1148. 2019. doi:10.3390/math7121148.
* "Definite Integral of Arctangent and Polylogarithmic Functions Expressed as a Series". Mathematics 7 (1099): 1099. 2019. doi:10.3390/math7111099.
* "Eigenschaften Einiger Bestimmten Integrale". Hof, K.K., Ed.. 1861.
* Mathematical handbook of formulas and tables (3rd ed.). McGraw-Hill. 2009. ISBN 978-0071548557.
* CRC standard mathematical tables and formulae (32nd ed.). CRC Press. 2003. ISBN 978-143983548-7.
* Abramowitz, Milton; Stegun, Irene Ann, eds (1983). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. LCCN 65-12253. ISBN 978-0-486-61272-0.

* v
* t
* e

Lists of integrals

* Rational functions
* Irrational functions
* Trigonometric functions
* Inverse trigonometric functions
* Hyperbolic functions
* Inverse hyperbolic functions
* Exponential functions
* Logarithmic functions
* Gaussian functions

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/List of definite integrals. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=List_of_definite_integrals&oldid=4350132"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template