List of definite integrals

In mathematics, the definite integral:

\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:

\int_a^\infty f(x)\, dx=\lim_{b \to \infty } \int_a^b f(x)\, dx

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

Definite integrals involving rational or irrational expression

\int_0^\infty \frac{dx}{x^{2}+a^{2}}=\frac{\pi }{2a}
\int_0^\infty \frac{x^{m}dx}{x^{n}+a^{n}}=\frac{\pi a^{m-n+1}}{n \sin [(m+1)(\pi /n)]} ,0<m+1<n
\int_0^\infty \frac{x^{p-1}dx}{1+x}= \frac{\pi }{\sin p\pi } \ \ , 0<p<1
\int_0^\infty \frac{x^{m}dx}{1+2x\cos\beta +x^{2}}=\frac{\pi }{\sin (m\pi) }\frac {\sin (m\beta)}{\sin \beta}
\int_0^\infty \frac{dx}{\sqrt{a^{2}-x^{2}}}=\frac{\pi }{2}
\int_0^a \sqrt{a^{2}-x^{2}} \, dx =\frac{\pi a^2}{4}
\int_0^a x^m (a^n-x^n)^p\,dx=\frac{a^{m+1+np}\Gamma [(m+1)/n]\Gamma(p+1)}{n\Gamma [((m+1)/n)+p+1]}
\int_0^\infty \frac{x^m \, dx}{({x^n+a^n)}^r}=\frac{(-1)^{r-1}\pi a^{m+1-nr}\Gamma [(m+1)/n]}{n\sin[(m+1)\pi/n](r-1)!\Gamma[(m+1)/n-r+1]} \ \ , n(r-2)<m+1<nr

Definite integrals involving trigonometric functions

\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} \ \ m,n \text{ positive integers}
\int_0^\pi \cos mx \cos nxdx=\begin{cases} 0 & \text{if } m\neq n \\ \dfrac{\pi}{2} & \text{if } m=n \end{cases} \ \ m,n \text{ positive integers}
\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} \ \ m,n \text{ integers}.
\int_0^{\frac{\pi}{2}}\sin^{2} x\, dx=\int_0^{\frac{\pi}{2}}\cos^{2} x\, dx=\frac{\pi}{4}
\int_0^{\frac{\pi}{2}}\sin^{2m} x\, dx=\int_0^{\frac{\pi}{2}}\cos^{2m} x\, dx = \frac{1\times3\times5\times\cdots\times(2m-1)}{2\times4\times6\times\cdots\times2m}\frac{\pi}{2} \ \ m=1,2,3,\ldots
\int_0^{\frac{\pi}{2}}\sin^{2m+1} x\, dx=\int_0^{\frac{\pi}{2}}\cos^{2m+1} x\, dx = \frac{2\times4\times6\times\cdots\times2m}{1\times3\times5\times\cdots\times(2m+1)} \ \ m=1,2,3,\ldots
\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)
\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)
\int_{0}^{\infty }\frac{\sin px\cos qx}{x}\ dx=\begin{cases} 0 & \text{ if } p>q>0 \\ \\ \dfrac{\pi}{2}& \text{ if } 0<p<q \\ \\ \dfrac{\pi}{4} & \text{ if } p=q>0 \end{cases}
\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}
\int_{0}^{\infty} \frac{\sin ^{2}px}{x^{2}}\ dx=\frac{\pi p}{2}
\int_{0}^{\infty} \frac{1-\cos px}{x^{2}}\ dx=\frac{\pi p}{2}
\int_{0}^{\infty} \frac{\cos px - \cos qx}{x}\ dx= \ln \frac {q}{p}
\int_{0}^{\infty} \frac{\cos px - \cos qx}{x^{2}}\ dx=\frac{\pi (q-p)}{2}
\int_{0}^{\infty} \frac{\cos mx}{x^{2}+a^{2}}\ dx=\frac{\pi}{2a}e^{-ma}
\int_0^\infty \frac{x \sin mx}{x^2+a^2}\ dx=\frac{\pi}{2}e^{-ma}
\int_0^\infty \frac{ \sin mx}{x(x^2+a^2)}\ dx=\frac{\pi}{2a^2}(1-e^{-ma})
\int_0^{2\pi} \frac{dx}{a+b\sin x}=\frac{2\pi}{\sqrt{a^2-b^2}}
\int_0^{2\pi} \frac{dx}{a+b\cos x}=\frac{2\pi}{\sqrt{a^2-b^2}}
\int_0^{\frac{\pi}{2}} \frac{dx}{a+b\cos x}=\frac{\cos^{-1}(b/a)}{\sqrt{a^2-b^2}}
\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}}
\int_0^{2\pi} \frac{dx}{1-2a\cos x +a^2}=\frac{2\pi}{1-a^2}\ \ \ ,\ 0<a<1
\int_0^{\pi} \frac{x \sin x\ dx}{1-2a\cos x +a^2}=\begin{cases} \frac{\pi}{a}\ln\left|1+a\right| & \text{if } |a|<1 \\ \frac{\pi}{a} \ln\left|1+\frac{1}{a}\right| & \text{if } |a|>1 \end{cases}
\int_0^{\pi} \frac{\cos mx\ dx}{1-2a\cos x +a^2}=\frac{\pi a^m}{1-a^2} \quad , a^2<1, \ m=0,1,2,\dots
\int_0^\infty \sin ax^2\ dx=\int_0^\infty \cos ax^2= \frac{1}{2}\sqrt \frac{\pi}{2a}
\int_0^\infty \sin ax^n=\frac{1}{na^{1/n}}\Gamma(1/n)\sin\frac{\pi}{2n}\quad ,n>1
\int_0^\infty \cos ax^n=\frac{1}{na^{1/n}}\Gamma(1/n)\cos\frac{\pi}{2n}\quad ,n>1
\int_0^\infty \frac{\sin x}{\sqrt x}\ dx=\int_0^\infty \frac{\cos x}{\sqrt x}\ dx=\sqrt{\frac{\pi}{2}}
\int_0^\infty \frac{\sin x}{x^p}\ dx= \frac{\pi}{2\Gamma(p)\sin (p\pi/2)}, \quad 0<p<1
\int_0^\infty \frac{\cos x}{x^p}\ dx= \frac{\pi}{2\Gamma(p)\cos (p\pi/2)}, \quad 0<p<1
\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)
\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)

Definite integrals involving exponential functions

\int_0^\infty e^{-ax}\cos bx \, dx=\frac{a}{a^2+b^2}
\int_0^\infty e^{-ax}\sin bx \, dx=\frac{b}{a^{2}+b^{2}}
\int_0^\infty \frac {{}e^{-ax}\sin bx}{x} \, dx=\tan^{-1}\frac{b}{a}
\int_0^\infty \frac {e^{-ax}-e^{-bx}}{x} \, dx=\ln \frac{b}{a}
\int_0^\infty {e^{-ax^{2}}}\, dx=\frac {1}{2} \sqrt{\frac {\pi}{a}}
\int_0^\infty {e^{-ax^{2}}}\cos bx\, dx=\frac {1}{2} \sqrt{\frac{\pi}{a}}e^{-b^{2}/4a}
\int_0^\infty e^{-(ax^{2}+bx+c)}\, dx=\frac{1}{2}\sqrt{\frac{\pi}{a}}e^{(b^{2}-4ac)/4a}\ \operatorname{erfc} \frac{b}{2\sqrt{a}},\text{ where }\operatorname{erfc}(p)=\frac{2}{\sqrt{\pi}}\int_p^\infty e^{-x^{2}}\, dx
\int_{-\infty}^{+\infty} e^{-(ax^{2}+bx+c)}\ dx=\sqrt {\frac{\pi}{a}}e^{(b^{2}-4ac)/4a}
\int_0^\infty x^{n}e^{-ax}\ dx=\frac{\Gamma (n+1)}{a^{n+1}}
\int_0^\infty x^{m}e^{-ax^2}\ dx=\frac{\Gamma [(m+1)/2]}{2a^{(m+1)/2}}
\int_0^\infty e^{-ax^{2}-b/x^{2}}\ dx=\frac{1}{2} \sqrt \frac{\pi}{a}e^{-2 \sqrt{ab}}
\int_0^\infty \frac {x}{e^{x}-1}\ dx=\zeta (2)= \frac {\pi^2}{6}
\int_0^\infty \frac {x^{n-1}}{e^{x}-1}\ dx=\Gamma (n)\zeta (n)
\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}
\int_0^\infty \frac {\sin mx}{e^{2\pi x}-1}\ dx=\frac{1}{4} \coth\frac{m}{2}- \frac{1}{2m}
\int_0^\infty \left(\frac {1}{1+x}- e^{-x}\right)\ \frac{dx}{x}=\gamma
\int_0^\infty \frac {e^{-x^2}-e^{-x}}{x}\ dx=\frac{\gamma}{2}
\int_0^\infty \left(\frac {1}{e^x-1}-\frac{e^{-x}}{x}\right)\ dx=\gamma
\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}
\int_0^\infty \frac {e^{-ax}-e^{-bx}}{x \csc px}\ dx=\tan^{-1}\frac{b}{p}-\tan^{-1}\frac{a}{p}
\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|
\int_{-\infty}^\infty e^{-x^2}\,dx=\sqrt{\pi}
\int_{-\infty}^\infty x^{2(n+1)}e^{-x^{2}/2}\,dx=\frac{(2n+1)!}{2^{n}n!}\sqrt{2 \pi} \quad n=0,1,2,\ldots

Definite integrals involving logarithmic functions

\int_0^1 x^m (\ln x)^n \, dx=\frac{(-1)^n n!}{(m+1)^{n+1}} \quad m>-1, n=0,1,2,\ldots
\int_0^1 \frac{\ln x}{1+x}\, dx= -\frac{\pi^2}{12}
\int_0^1 \frac{\ln x}{1-x}\, dx= -\frac{\pi^2}{6}
\int_0^1 \frac{\ln (1+x)}{x}\, dx= \frac{\pi^2}{12}
\int_0^1 \frac{\ln (1-x)}{x}\, dx= -\frac{\pi^2}{6}
\int_{0}^{\infty} \frac{\ln(a^{2}+x^{2})}{b^{2}+x^{2}}\ dx = \frac{\pi}{b} \ln (a+b)\quad a,b>0
\int_{0}^{\infty}\frac{\ln x}{x^2+a^2}\ dx = \frac{\pi \ln a}{2a}\quad a>0

Definite integrals involving hyperbolic functions

\int_{0}^{\infty }\frac{\sin ax}{\sinh bx}\ dx=\frac {\pi}{2b}\tanh \frac{a \pi}{2b}

\int_{0}^{\infty }\frac{\cos ax}{\cosh bx}\ dx=\frac {\pi}{2b}\frac{1}{\cosh \frac{a \pi}{2b}}

\int_{0}^{\infty }\frac{x}{\sinh ax}\ dx=\frac{\pi^{2}}{4a^{2}}

\int_{-\infty}^{\infty}\frac{1}{\cosh x}\ dx = \pi

Miscellaneous definite integrals

\int_{0}^{\infty }\frac{f(ax)-f(bx)}{x}\ dx=[{f(0)-f(\infty)}]\ln \frac{b}{a}

See also

References

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  1. 1 2 3 Zwillinger, Daniel (2003). CRC standard mathematical tables and formulae (32nd ed.). CRC Press. ISBN 978-1439835487.
  2. 1 2 3 Abramowitz, Milton; Stegun, Irene A. (1965). Handbook of mathematical functions with formulas, graphs, and mathematical tables (Unabridged and unaltered republ. [der Ausg.] 1964, 5. Dover printing ed.). U.S. Govt. Print. Off. ISBN 978-0486612720.
  3. 1 2 Murray R. Spiegel, Seymour Lipschutz, John Liu (2009). Mathematical handbook of formulas and tables (3rd ed.). McGraw-Hill. ISBN 978-0071548557.
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