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:

$\int_a^\infty f(x)dx=\lim_{b\to\infty}\left[\int_a^b f(x)dx\right]$

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

$\int_0^\infty \frac{dx}{x^{2}+a^{2}}=\frac{\pi}{2a}$

$\int_0^\infty \frac{x^{m}dx}{x^{n}+a^{n}}=\frac{\pi a^{m-n+1}}{n\sin\left(\frac{(m+1)\pi}{n}\right)}\ ,\ 0<m+1<n$

$\int_0^\infty \frac{x^{p-1}dx}{1+x}=\frac{\pi}{\sin(p\pi)}\ ,\ 0<p<1$

$\int_0^\infty \frac{x^m dx}{1+2x\cos\beta+x^{2}}=\frac{\pi}{\sin(m\pi)}\frac{\sin(m\beta)}{\sin(\beta)}$

$\int_0^\infty \frac{dx}{\sqrt{a^2-x^2}}=\frac{\pi}{2}$

$\int_0^a \sqrt{a^2-x^2}dx=\frac{\pi a^2}{4}$

$\int_0^a x^m (a^n-x^n)^p\,dx=\frac{a^{m+1+np}\Gamma [(m+1)/n]\Gamma(p+1)}{n\Gamma [((m+1)/n)+p+1]}$

$\int_0^\infty \frac{x^m dx}{({x^n+a^n)}^r}=\frac{(-1)^{r-1}\pi a^{m+1-nr}\Gamma [(m+1)/n]}{n\sin[(m+1)\pi/n](r-1)!\Gamma[(m+1)/n-r+1]}\ ,\ n(r-2)<m+1<nr$

Definite integrals involving trigonometric functions

$\int_0^\pi \sin(mx)\sin(nx)dx=\begin{cases} 0 & \text{if } m\ne n \\ \dfrac{\pi}{2} & \text{if } m=n \end{cases} \ \ m,n \text{ positive integers}$

$\int_0^\pi \cos(mx)\cos(nx)dx=\begin{cases} 0 & \text{if } m\ne n \\ \dfrac{\pi}{2} & \text{if } m=n \end{cases} \ \ m,n \text{ positive integers}$

$\int_0^\pi \sin(mx)\cos(nx)dx=\begin{cases} 0 & \text{if } m+n \text{ even} \\ \dfrac{2m}{m^{2}-n^{2}} & \text{if } m+n \text{ odd} \end{cases} \ \ m,n \text{ integers}.$

$\int_0^{\frac{\pi}{2}}\sin^2(x)dx=\int_0^{\frac{\pi}{2}}\cos^2(x)dx=\frac{\pi}{4}$

$\int_0^{\frac{\pi}{2}}\sin^{2m}(x)dx=\int_0^{\frac{\pi}{2}}\cos^{2m}(x)dx = \frac{1\times3\times5\times\cdots\times(2m-1)}{2\times4\times6\times\cdots\times2m}\frac{\pi}{2} \ \ m=1,2,3,\ldots$

$\int_0^{\frac{\pi}{2}}\sin^{2m+1}(x)dx=\int_0^{\frac{\pi}{2}}\cos^{2m+1}(x)dx = \frac{2\times4\times6\times\cdots\times2m}{1\times3\times5\times\cdots\times(2m+1)} \ \ m=1,2,3,\ldots$

$\int_0^{\frac{\pi}{2}}\sin^{2p-1}(x)\cos^{2q-1}(x)dx = \frac{\Gamma(p)\Gamma(q)}{2\Gamma(p+q)}=\frac{1}{2} \text{B}(p,q)$

$\int_0^\infty \frac{\sin(px)}{x}dx=\begin{cases} \dfrac{\pi}{2} & \text{if } p>0 \\ \\ 0 & \text{if } p=0 \\ \\ -\dfrac{\pi}{2} & \text {if } p<0 \end{cases}$ (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

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

References

• 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 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C., USA; New York, USA: United States Department of Commerce, National Bureau of Standards; Dover Publications. ISBN 0-486-61272-4. LCCN 64-60036. MR 0167642. ISBN 978-0-486-61272-0. LCCN 65-12253.