List of definite integrals

Part of a series of articles about
Calculus
${\displaystyle \int _{a}^{b}f'(t)\,dt=f(b)-f(a)}$
Fundamental theorem Limits Continuity Rolle's theorem Mean value theorem Inverse function theorem
Differential Definitions Derivative (generalizations) Differential infinitesimal of a function total Concepts Differentiation notation Second derivative Implicit differentiation Logarithmic differentiation Related rates Taylor's theorem Rules and identities Sum Product Chain Power Quotient L'Hôpital's rule Inverse General Leibniz Faà di Bruno's formula Reynolds
Integral Lists of integrals Integral transform Leibniz integral rule Definitions Antiderivative Integral (improper) Riemann integral Lebesgue integration Contour integration Integral of inverse functions Integration by Parts Discs Cylindrical shells Substitution (trigonometric, tangent half-angle, Euler) Euler's formula Partial fractions (Heaviside's method) Changing order Reduction formulae Differentiating under the integral sign Risch algorithm
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 Gradient Green's Stokes' Divergence generalized Stokes Helmholtz decomposition
Multivariable Formalisms Matrix Tensor Exterior Geometric Definitions Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian Hessian
Advanced Calculus on Euclidean space Generalized functions Limit of distributions
Specialized Fractional Malliavin Stochastic Variations
Miscellanea Precalculus History Glossary List of topics Integration Bee Mathematical analysis Nonstandard analysis
v t e

In mathematics, the definite integral

${\displaystyle \int _{a}^{b}f(x)\,dx}$

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:

${\displaystyle \int _{a}^{\infty }f(x)\,dx=\lim _{b\to \infty }\left[\int _{a}^{b}f(x)\,dx\right]}$

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.

${\displaystyle \int _{0}^{\infty }{\frac {dx}{1+x^{p}={\frac {\pi /p}{\sin(\pi /p)}\quad {\text{for }\Re (p)>1}$

${\displaystyle \int _{0}^{\infty }{\frac {x^{p-1}dx}{1+x}={\frac {\pi }{\sin(p\pi )}\quad {\text{for }0<p<1}$

${\displaystyle \int _{0}^{\infty }{\frac {x^{m}dx}{x^{n}+a^{n}={\frac {\pi a^{m-n+1}{n\sin \left({\dfrac {m+1}{n}\pi \right)}\quad {\text{for }0<m+1<n}$

${\displaystyle \int _{0}^{\infty }{\frac {x^{m}dx}{1+2x\cos \beta +x^{2}={\frac {\pi }{\sin(m\pi )}\cdot {\frac {\sin(m\beta )}{\sin(\beta )}$

${\displaystyle \int _{0}^{a}{\frac {dx}{\sqrt {a^{2}-x^{2}={\frac {\pi }{2}$

${\displaystyle \int _{0}^{a}{\sqrt {a^{2}-x^{2}dx={\frac {\pi a^{2}{4}$

${\displaystyle \int _{0}^{a}x^{m}(a^{n}-x^{n})^{p}\,dx={\frac {a^{m+1+np}\Gamma \left({\dfrac {m+1}{n}\right)\Gamma (p+1)}{n\Gamma \left({\dfrac {m+1}{n}+p+1\right)}$

${\displaystyle \int _{0}^{\infty }{\frac {x^{m}dx}{({x^{n}+a^{n})}^{r}={\frac {(-1)^{r-1}\pi a^{m+1-nr}\Gamma \left({\dfrac {m+1}{n}\right)}{n\sin \left({\dfrac {m+1}{n}\pi \right)(r-1)!\,\Gamma \left({\dfrac {m+1}{n}-r+1\right)}\quad {\text{for }n(r-2)<m+1<nr}$

${\displaystyle \int _{0}^{\pi }\sin(mx)\sin(nx)dx={\begin{cases}0&{\text{if }m\neq n\\\\{\dfrac {\pi }{2}&{\text{if }m=n\end{cases}\quad {\text{for }m,n{\text{ positive integers}$

${\displaystyle \int _{0}^{\pi }\cos(mx)\cos(nx)dx={\begin{cases}0&{\text{if }m\neq n\\\\{\dfrac {\pi }{2}&{\text{if }m=n\end{cases}\quad {\text{for }m,n{\text{ positive integers}$

${\displaystyle \int _{0}^{\pi }\sin(mx)\cos(nx)dx={\begin{cases}0&{\text{if }m+n{\text{ even}\\\\{\dfrac {2m}{m^{2}-n^{2}&{\text{if }m+n{\text{ odd}\end{cases}\quad {\text{for }m,n{\text{ integers}.}$

${\displaystyle \int _{0}^{\frac {\pi }{2}\sin ^{2}(x)dx=\int _{0}^{\frac {\pi }{2}\cos ^{2}(x)dx={\frac {\pi }{4}$

${\displaystyle \int _{0}^{\frac {\pi }{2}\sin ^{2m}(x)dx=\int _{0}^{\frac {\pi }{2}\cos ^{2m}(x)dx={\frac {1\times 3\times 5\times \cdots \times (2m-1)}{2\times 4\times 6\times \cdots \times 2m}\cdot {\frac {\pi }{2}\quad {\text{for }m=1,2,3\ldots }$

${\displaystyle \int _{0}^{x}\sin ^{2m}(t)dt={\frac {(2m-1)!!}{(2m)!!}{\Biggl (}x-\sin(x)\cos(x){\Biggl (}1+\sum _{k=1}^{\infty }{\frac {\sin ^{2k}(x)(2k)!!}{(2k+1)!!}{\Biggr )}{\Biggr )}\quad {\text{for }m=1,2,3\ldots }$

${\displaystyle \int _{0}^{x}\cos ^{2m}(t)dt={\frac {(2m-1)!!}{(2m)!!}{\Biggl (}x-\sin(x)\cos(x){\Biggl (}1+\sum _{k=1}^{\infty }{\frac {\cos ^{2k}(x)(2k)!!}{(2k+1)!!}{\Biggr )}{\Biggr )}\quad {\text{for }m=1,2,3\ldots }$

${\displaystyle \int _{0}^{\frac {\pi }{2}\sin ^{2m+1}(x)dx=\int _{0}^{\frac {\pi }{2}\cos ^{2m+1}(x)dx={\frac {2\times 4\times 6\times \cdots \times 2m}{1\times 3\times 5\times \cdots \times (2m+1)}\quad {\text{for }m=1,2,3\ldots }$

${\displaystyle \int _{0}^{\frac {\pi }{2}\sin ^{2p-1}(x)\cos ^{2q-1}(x)dx={\frac {\Gamma (p)\Gamma (q)}{2\Gamma (p+q)}={\frac {1}{2}{\text{B}(p,q)}$

${\displaystyle \int _{0}^{\infty }{\frac {\sin(px)}{x}dx={\begin{cases}{\dfrac {\pi }{2}&{\text{if }p>0\\\\0&{\text{if }p=0\\\\-{\dfrac {\pi }{2}&{\text{if }p<0\end{cases}$ (see Dirichlet integral)

${\displaystyle \int _{0}^{\infty }{\frac {\sin px\cos qx}{x}\ dx={\begin{cases}0&{\text{ if }q>p>0\\\\{\dfrac {\pi }{2}&{\text{ if }0<q<p\\\\{\dfrac {\pi }{4}&{\text{ if }p=q>0\end{cases}$

${\displaystyle \int _{0}^{\infty }{\frac {\sin px\sin qx}{x^{2}\ dx={\begin{cases}{\dfrac {\pi p}{2}&{\text{ if }0<p\leq q\\\\{\dfrac {\pi q}{2}&{\text{ if }0<q\leq p\end{cases}$

${\displaystyle \int _{0}^{\infty }{\frac {\sin ^{2}px}{x^{2}\ dx={\frac {\pi p}{2}$

${\displaystyle \int _{0}^{\infty }{\frac {1-\cos px}{x^{2}\ dx={\frac {\pi p}{2}$

${\displaystyle \int _{0}^{\infty }{\frac {\cos px-\cos qx}{x}\ dx=\ln {\frac {q}{p}$

${\displaystyle \int _{0}^{\infty }{\frac {\cos px-\cos qx}{x^{2}\ dx={\frac {\pi (q-p)}{2}$

${\displaystyle \int _{0}^{\infty }{\frac {\cos mx}{x^{2}+a^{2}\ dx={\frac {\pi }{2a}e^{-ma}$

${\displaystyle \int _{0}^{\infty }{\frac {x\sin mx}{x^{2}+a^{2}\ dx={\frac {\pi }{2}e^{-ma}$

${\displaystyle \int _{0}^{\infty }{\frac {\sin mx}{x(x^{2}+a^{2})}\ dx={\frac {\pi }{2a^{2}\left(1-e^{-ma}\right)}$

${\displaystyle \int _{0}^{2\pi }{\frac {dx}{a+b\sin x}={\frac {2\pi }{\sqrt {a^{2}-b^{2}$

${\displaystyle \int _{0}^{2\pi }{\frac {dx}{a+b\cos x}={\frac {2\pi }{\sqrt {a^{2}-b^{2}$

${\displaystyle \int _{0}^{\frac {\pi }{2}{\frac {dx}{a+b\cos x}={\frac {\cos ^{-1}\left({\dfrac {b}{a}\right)}{\sqrt {a^{2}-b^{2}$

${\displaystyle \int _{0}^{2\pi }{\frac {dx}{(a+b\sin x)^{2}=\int _{0}^{2\pi }{\frac {dx}{(a+b\cos x)^{2}={\frac {2\pi a}{(a^{2}-b^{2})^{3/2}$

${\displaystyle \int _{0}^{2\pi }{\frac {dx}{1-2a\cos x+a^{2}={\frac {2\pi }{1-a^{2}\quad {\text{for }0<a<1}$

${\displaystyle \int _{0}^{\pi }{\frac {x\sin x\ dx}{1-2a\cos x+a^{2}={\begin{cases}{\dfrac {\pi }{a}\ln \left|1+a\right|&{\text{if }|a|<1\\\\{\dfrac {\pi }{a}\ln \left|1+{\dfrac {1}{a}\right|&{\text{if }|a|>1\end{cases}$

${\displaystyle \int _{0}^{\pi }{\frac {\cos mx\ dx}{1-2a\cos x+a^{2}={\frac {\pi a^{m}{1-a^{2}\quad {\text{for }a^{2}<1\ ,\ m=0,1,2,\dots }$

${\displaystyle \int _{0}^{\infty }\sin ax^{2}\ dx=\int _{0}^{\infty }\cos ax^{2}={\frac {1}{2}{\sqrt {\frac {\pi }{2a}$

${\displaystyle \int _{0}^{\infty }\sin ax^{n}={\frac {1}{na^{1/n}\Gamma \left({\frac {1}{n}\right)\sin {\frac {\pi }{2n}\quad {\text{for }n>1}$

${\displaystyle \int _{0}^{\infty }\cos ax^{n}={\frac {1}{na^{1/n}\Gamma \left({\frac {1}{n}\right)\cos {\frac {\pi }{2n}\quad {\text{for }n>1}$

${\displaystyle \int _{0}^{\infty }{\frac {\sin x}{\sqrt {x}\ dx=\int _{0}^{\infty }{\frac {\cos x}{\sqrt {x}\ dx={\sqrt {\frac {\pi }{2}$

${\displaystyle \int _{0}^{\infty }{\frac {\sin x}{x^{p}\ dx={\frac {\pi }{2\Gamma (p)\sin \left({\dfrac {p\pi }{2}\right)}\quad {\text{for }0<p<1}$

${\displaystyle \int _{0}^{\infty }{\frac {\cos x}{x^{p}\ dx={\frac {\pi }{2\Gamma (p)\cos \left({\dfrac {p\pi }{2}\right)}\quad {\text{for }0<p<1}$

${\displaystyle \int _{0}^{\infty }\sin ax^{2}\cos 2bx\ dx={\frac {1}{2}{\sqrt {\frac {\pi }{2a}\left(\cos {\frac {b^{2}{a}-\sin {\frac {b^{2}{a}\right)}$

${\displaystyle \int _{0}^{\infty }\cos ax^{2}\cos 2bx\ dx={\frac {1}{2}{\sqrt {\frac {\pi }{2a}\left(\cos {\frac {b^{2}{a}+\sin {\frac {b^{2}{a}\right)}$

${\displaystyle \int _{0}^{\infty }{\sqrt {x}\,e^{-x}\,dx={\frac {1}{2}{\sqrt {\pi }$ (see also Gamma function)

${\displaystyle \int _{0}^{\infty }e^{-ax}\cos bx\,dx={\frac {a}{a^{2}+b^{2}$

${\displaystyle \int _{0}^{\infty }e^{-ax}\sin bx\,dx={\frac {b}{a^{2}+b^{2}$

${\displaystyle \int _{0}^{\infty }{\frac {}e^{-ax}\sin bx}{x}\,dx=\tan ^{-1}{\frac {b}{a}$

${\displaystyle \int _{0}^{\infty }{\frac {e^{-ax}-e^{-bx}{x}\,dx=\ln {\frac {b}{a}$

${\displaystyle \int _{0}^{\infty }{\frac {e^{-ax}-\cos(bx)}{x}\,dx=\ln {\frac {b}{a}$

${\displaystyle \int _{0}^{\infty }e^{-ax^{2}\,dx={\frac {1}{2}{\sqrt {\frac {\pi }{a}\quad {\text{for }a>0}$ (the Gaussian integral)

${\displaystyle \int _{0}^{\infty }{e^{-ax^{2}\cos bx\,dx={\frac {1}{2}{\sqrt {\frac {\pi }{a}e^{\left({\frac {-b^{2}{4a}\right)}$

${\displaystyle \int _{0}^{\infty }e^{-(ax^{2}+bx+c)}\,dx={\frac {1}{2}{\sqrt {\frac {\pi }{a}e^{\left({\frac {b^{2}-4ac}{4a}\right)}\cdot \operatorname {erfc} {\frac {b}{2{\sqrt {a},{\text{ where }\operatorname {erfc} (p)={\frac {2}{\sqrt {\pi }\int _{p}^{\infty }e^{-x^{2}\,dx}$

${\displaystyle \int _{-\infty }^{\infty }e^{-(ax^{2}+bx+c)}\ dx={\sqrt {\frac {\pi }{a}e^{\left({\frac {b^{2}-4ac}{4a}\right)}$

${\displaystyle \int _{0}^{\infty }x^{n}e^{-ax}\ dx={\frac {\Gamma (n+1)}{a^{n+1}$

${\displaystyle \int _{0}^{\infty }{x^{2}e^{-ax^{2}\,dx}={\frac {1}{4}{\sqrt {\frac {\pi }{a^{3}\quad {\text{for }a>0}$

${\displaystyle \int _{0}^{\infty }x^{2n}e^{-ax^{2}\,dx={\frac {2n-1}{2a}\int _{0}^{\infty }x^{2(n-1)}e^{-ax^{2}\,dx={\frac {(2n-1)!!}{2^{n+1}{\sqrt {\frac {\pi }{a^{2n+1}={\frac {(2n)!}{n!2^{2n+1}{\sqrt {\frac {\pi }{a^{2n+1}\quad {\text{for }a>0\ ,\ n=1,2,3\ldots }$ (where !! is the double factorial)

${\displaystyle \int _{0}^{\infty }{x^{3}e^{-ax^{2}\,dx}={\frac {1}{2a^{2}\quad {\text{for }a>0}$

${\displaystyle \int _{0}^{\infty }x^{2n+1}e^{-ax^{2}\,dx={\frac {n}{a}\int _{0}^{\infty }x^{2n-1}e^{-ax^{2}\,dx={\frac {n!}{2a^{n+1}\quad {\text{for }a>0\ ,\ n=0,1,2\ldots }$

${\displaystyle \int _{0}^{\infty }x^{m}e^{-ax^{2}\ dx={\frac {\Gamma \left({\dfrac {m+1}{2}\right)}{2a^{\left({\frac {m+1}{2}\right)}$

${\displaystyle \int _{0}^{\infty }e^{\left(-ax^{2}-{\frac {b}{x^{2}\right)}\ dx={\frac {1}{2}{\sqrt {\frac {\pi }{a}e^{-2{\sqrt {ab}$

${\displaystyle \int _{0}^{\infty }{\frac {x}{e^{x}-1}\ dx=\zeta (2)={\frac {\pi ^{2}{6}$

${\displaystyle \int _{0}^{\infty }{\frac {x^{n-1}{e^{x}-1}\ dx=\Gamma (n)\zeta (n)}$

${\displaystyle \int _{0}^{\infty }{\frac {x}{e^{x}+1}\ dx={\frac {1}{1^{2}-{\frac {1}{2^{2}+{\frac {1}{3^{2}-{\frac {1}{4^{2}+\dots ={\frac {\pi ^{2}{12}$

${\displaystyle \int _{0}^{\infty }{\frac {x^{n}{e^{x}+1}\ dx=n!\cdot \left({\frac {2^{n}-1}{2^{n}\right)\zeta (n+1)}$

${\displaystyle \int _{0}^{\infty }{\frac {\sin mx}{e^{2\pi x}-1}\ dx={\frac {1}{4}\coth {\frac {m}{2}-{\frac {1}{2m}$

${\displaystyle \int _{0}^{\infty }\left({\frac {1}{1+x}-e^{-x}\right)\ {\frac {dx}{x}=\gamma }$ (where ${\displaystyle \gamma }$ is Euler–Mascheroni constant)

${\displaystyle \int _{0}^{\infty }{\frac {e^{-x^{2}-e^{-x}{x}\ dx={\frac {\gamma }{2}$

${\displaystyle \int _{0}^{\infty }\left({\frac {1}{e^{x}-1}-{\frac {e^{-x}{x}\right)\ dx=\gamma }$

${\displaystyle \int _{0}^{\infty }{\frac {e^{-ax}-e^{-bx}{x\sec px}\ dx={\frac {1}{2}\ln {\frac {b^{2}+p^{2}{a^{2}+p^{2}$

${\displaystyle \int _{0}^{\infty }{\frac {e^{-ax}-e^{-bx}{x\csc px}\ dx=\tan ^{-1}{\frac {b}{p}-\tan ^{-1}{\frac {a}{p}$

${\displaystyle \int _{0}^{\infty }{\frac {e^{-ax}(1-\cos x)}{x^{2}\ dx=\cot ^{-1}a-{\frac {a}{2}\ln \left|{\frac {a^{2}+1}{a^{2}\right|}$

${\displaystyle \int _{-\infty }^{\infty }e^{-x^{2}\,dx={\sqrt {\pi }$

${\displaystyle \int _{-\infty }^{\infty }x^{2(n+1)}e^{-{\frac {1}{2}x^{2}\,dx={\frac {(2n+1)!}{2^{n}n!}{\sqrt {2\pi }\quad {\text{for }n=0,1,2,\ldots }$

${\displaystyle \int _{0}^{1}x^{m}(\ln x)^{n}\,dx={\frac {(-1)^{n}n!}{(m+1)^{n+1}\quad {\text{for }m>-1,n=0,1,2,\ldots }$

${\displaystyle \int _{1}^{\infty }x^{m}(\ln x)^{n}\,dx={\frac {(-1)^{n+1}n!}{(m+1)^{n+1}\quad {\text{for }m<-1,n=0,1,2,\ldots }$

${\displaystyle \int _{0}^{1}{\frac {\ln x}{1+x}\,dx=-{\frac {\pi ^{2}{12}$

${\displaystyle \int _{0}^{1}{\frac {\ln x}{1-x}\,dx=-{\frac {\pi ^{2}{6}$

${\displaystyle \int _{0}^{1}{\frac {\ln(1+x)}{x}\,dx={\frac {\pi ^{2}{12}$

${\displaystyle \int _{0}^{1}{\frac {\ln(1-x)}{x}\,dx=-{\frac {\pi ^{2}{6}$

${\displaystyle \int _{0}^{\infty }{\frac {\ln(a^{2}+x^{2})}{b^{2}+x^{2}\ dx={\frac {\pi }{b}\ln(a+b)\quad {\text{for }a,b>0}$

${\displaystyle \int _{0}^{\infty }{\frac {\ln x}{x^{2}+a^{2}\ dx={\frac {\pi \ln a}{2a}\quad {\text{for }a>0}$

$${\displaystyle \int _{0}^{\infty }{\frac {\sin ax}{\sinh bx}\ dx={\frac {\pi }{2b}\tanh {\frac {a\pi }{2b}$$

$${\displaystyle \int _{0}^{\infty }{\frac {\cos ax}{\cosh bx}\ dx={\frac {\pi }{2b}\cdot {\frac {1}{\cosh {\frac {a\pi }{2b}$$

$${\displaystyle \int _{0}^{\infty }{\frac {x}{\sinh ax}\ dx={\frac {\pi ^{2}{4a^{2}$$

$${\displaystyle \int _{0}^{\infty }{\frac {x^{2n+1}{\sinh ax}\ dx=c_{2n+1}\left({\frac {\pi }{a}\right)^{2(n+1)},\quad c_{2n+1}={\frac {(-1)^{n}{2}\left({\frac {1}{2}-\sum _{k=0}^{n-1}(-1)^{k}{2n+1 \choose 2k+1}c_{2k+1}\right),\quad c_{1}={\frac {1}{4}$$

$${\displaystyle \int _{0}^{\infty }{\frac {1}{\cosh ax}\ dx={\frac {\pi }{2a}$$

$${\displaystyle \int _{0}^{\infty }{\frac {x^{2n}{\cosh ax}\ dx=d_{2n}\left({\frac {\pi }{a}\right)^{2n+1},\quad d_{2n}={\frac {(-1)^{n}{2}\left({\frac {1}{4^{n}-\sum _{k=0}^{n-1}(-1)^{k}{2n \choose 2k}d_{2k}\right),\quad d_{0}={\frac {1}{2}$$

$${\displaystyle \int _{0}^{\infty }{\frac {f(ax)-f(bx)}{x}\ dx=\left(\lim _{x\to 0}f(x)-\lim _{x\to \infty }f(x)\right)\ln \left({\frac {b}{a}\right)}$$ holds if the integral exists and ${\displaystyle f'(x)}$ is continuous.

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

• Reynolds, Robert; Stauffer, Allan (2020). "Derivation of Logarithmic and Logarithmic Hyperbolic Tangent Integrals Expressed in Terms of Special Functions". Mathematics. 8 (687): 687. doi:10.3390/math8050687.
• Reynolds, Robert; Stauffer, Allan (2019). "A Definite Integral Involving the Logarithmic Function in Terms of the Lerch Function". Mathematics. 7 (1148): 1148. doi:10.3390/math7121148.
• Reynolds, Robert; Stauffer, Allan (2019). "Definite Integral of Arctangent and Polylogarithmic Functions Expressed as a Series". Mathematics. 7 (1099): 1099. doi:10.3390/math7111099.
• Winckler, Anton (1861). "Eigenschaften Einiger Bestimmten Integrale". Hof, K.K., Ed.
• Spiegel, Murray R.; Lipschutz, Seymour; Liu, John (2009). Mathematical handbook of formulas and tables (3rd ed.). McGraw-Hill. ISBN 978-0071548557.
• Zwillinger, Daniel (2003). CRC standard mathematical tables and formulae (32nd ed.). CRC Press. ISBN 978-143983548-7.
• Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 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. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253.