Penrose square root law

2013-12-06T12:26:43Z

195.110.141.124: /* Short derivation */ applications mentioned

{{calculus|expanded=Integral calculus}}

In mathematics, the [[definite integral]]:

: <math>\int_a^b f(x)\, dx</math>

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:

: <math>\int_a^\infty f(x)\, dx=\lim_{b \to \infty } \int_a^b f(x)\, dx </math>

The following is a list of the most common definite [[Integral]]s. For a list of [[indefinite integral]]s see [[List of integrals|''List of indefinite integrals'']]

==Definite integrals involving rational or irrational expression==

: <math> \int_0^\infty \frac{dx}{x^{2}+a^{2}}=\frac{\pi }{2a} </math>

: <math>\int_{0}^{a}\frac{x^{m}dx}{x^{n}+a^{n}}=\frac{\pi a^{m-n+1}}{n \sin [(m+1)\pi /n)]}</math>

: <math>\int_0^\infty \frac{x^{p-1}dx}{1+x}= \frac{\pi }{\sin p\pi } \ \ , 0<p<1</math>

: <math>\int_0^\infty \frac{x^{m}dx}{1+2x\cos\beta +x^{2}}=\frac{\pi }{\sin (m\pi) }\frac {\sin (m\beta)}{\sin \beta}</math>

: <math>\int_0^\infty \frac{dx}{\sqrt{a^{2}-x^{2}}}=\frac{\pi }{2} </math>

: <math>\int_0^a \sqrt{a^{2}-x^{2}} \, dx =\frac{\pi a^2}{4} </math>

: <math>\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]}</math>

: <math>\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</math>

==Definite integrals involving trigonometric functions==

: <math>\int_0^\pi \sin mx \sin nx\, dx=\begin{cases}
0 & \text{if } m\neq n \\
\pi/2 & \text{if } m=n
\end{cases}
\ \ m,n \text{ integers}</math>

: <math>\int_0^\pi \cos mx \cos nxdx=\begin{cases}
0 & \text{if } m\neq n \\
\pi/2 & \text{if } m=n
\end{cases}
\ \ m,n \text{ integers}</math>

: <math>\int_0^\pi \sin mx \cos nx\, dx=\begin{cases}
0 & \text{if } m+n \text{ even} \\
\frac{2m}{m^{2}-n^{2}} & \text{if } m+n \text{ odd}
\end{cases}
\ \ m,n \text{ integers}.</math>

: <math>\int_0^{\pi/2}\sin^{2} x\, dx=\int_0^{\pi/2}\cos^{2} x\, dx=\pi/4</math>

: <math>\int_0^{\pi/2}\sin^{2m} x\, dx=\int_0^{\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</math>

: <math>\int_0^{\pi/2}\sin^{2m+1} x\, dx=\int_0^{\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</math>

: <math>\int_0^{\pi/2}\sin^{2p-1} x \cos^{2q-1} x\, dx = \frac{\Gamma(p)\Gamma(q)}{2\Gamma(p+q)}=\frac{1}{2} B(p,q)</math>

: <math>\int_0^\infty \frac{\sin px}{x}\, dx=\begin{cases}
\pi/2 & \text{if } p>0 \\
0 & \text{if } p=0 \\
-\pi/2 & \text {if } p<0
\end{cases}</math>

<math>\int_{0}^{\infty }\frac{\sin px\cos qx}{x}\ dx=\begin{cases}
0 & \text{ if } p>q>0 \\
\pi/2& \text{ if } 0<p<q \\
\pi/4 & \text{ if } p=q>0
\end{cases}</math>

<math>\int_{0}^{\infty }\frac{\sin px \sin qx}{x^{2}}\ dx=\begin{cases}
\pi p/2& \text{ if } 0<p\leq q \\
\pi q/2 & \text{ if } 0<q\leq p
\end{cases}</math>

<math>\int_{0}^{\infty} \frac{\sin ^{2}px}{x^{2}}\ dx=\frac{\pi p}{2}</math>

<math>\int_{0}^{\infty} \frac{1-\cos px}{x^{2}}\ dx=\frac{\pi p}{2}</math>

<math>\int_{0}^{\infty} \frac{\cos px - \cos qx}{x}\ dx= \ln \frac {q}{p}</math>

<math>\int_{0}^{\infty} \frac{\cos px - \cos qx}{x^{2}}\ dx=\frac{\pi (q-p)}{2}</math>

<math>\int_{0}^{\infty} \frac{\cos mx}{x^{2}+a^{2}}\ dx=\frac{\pi}{2a}e^{-ma}</math>

: <math>\int_0^\infty \frac{x \sin mx}{x^2+a^2}\ dx=\frac{\pi}{2}e^{-ma}</math>

: <math>\int_0^\infty \frac{ \sin mx}{x(x^2+a^2)}\ dx=\frac{\pi}{2a^2}(1-e^{-ma})</math>

: <math>\int_0^{2\pi} \frac{dx}{a+b\sin x}=\frac{2\pi}{\sqrt{a^2-b^2}}</math>

: <math>\int_0^{2\pi} \frac{dx}{a+b\cos x}=\frac{2\pi}{\sqrt{a^2-b^2}}</math>

: <math>\int_0^{\frac{\pi}{2}} \frac{dx}{a+b\cos x}=\frac{\cos^{-1}(b/a)}{\sqrt{a^2-b^2}}</math>

: <math>\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}}</math>

: <math>\int_0^{2\pi} \frac{dx}{1-2a\cos x +a^2}=\frac{2\pi}{1-a^2}\ \ \ ,\ 0<a<1</math>

: <math>\int_0^{\pi} \frac{x \sin x\ dx}{1-2a\cos x +a^2}=\begin{cases}
\frac{\pi}{a}\ln (1+a) & \text{if } |a|<1 \\
\pi \ln(1+1/a) & \text{if } |a|>1
\end{cases}</math>

: <math>\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</math>

: <math>\int_0^\infty \sin ax^2\ dx=\int_0^\infty \cos ax^2= \frac{1}{2}\sqrt \frac{\pi}{2a}</math>

: <math>\int_0^\infty \sin ax^n=\frac{1}{na^{1/n}}\Gamma(1/n)\sin\frac{\pi}{2n}\quad ,n>1</math>

: <math>\int_0^\infty \cos ax^n=\frac{1}{na^{1/n}}\Gamma(1/n)\cos\frac{\pi}{2n}\quad ,n>1</math>

: <math>\int_0^\infty \frac{\sin x}{\sqrt x}\ dx=\int_0^\infty \frac{\cos x}{\sqrt x}\ dx=\sqrt{\frac{\pi}{2}}</math>

: <math>\int_0^\infty \frac{\sin x}{x^p}\ dx= \frac{\pi}{2\Gamma(p)\sin (p\pi/2)}, \quad 0<p<1</math>

: <math>\int_0^\infty \frac{\cos x}{x^p}\ dx= \frac{\pi}{2\Gamma(p)\cos (p\pi/2)}, \quad 0<p<1</math>

: <math>\int_0^\infty \sin ax^2\cos 2bx\ dx=\frac{1}{2}\sqrt{\frac{\pi}{2a}}(\cos \frac{b^2}{a}-\sin\frac{b^2}{a})</math>

: <math>\int_0^\infty \cos ax^2\cos 2bx\ dx=\frac{1}{2}\sqrt{\frac{\pi}{2a}}(\cos \frac{b^2}{a}+\sin\frac{b^2}{a})</math>

==Definite integrals involving exponential functions==

: <math>\int_0^\infty e^{-ax}\cos bx \, dx=\frac{a}{a^2+b^2}</math>

: <math>\int_0^\infty e^{-ax}\sin bx \, dx=\frac{b}{a^{2}+b^{2}}</math>

: <math>\int_0^\infty \frac {{}e^{-ax}\sin bx}{x} \, dx=\tan^{-1}\frac{b}{a}</math>

: <math>\int_0^\infty \frac {e^{-ax}-e^{-bx}}{x} \, dx=\ln \frac{b}{a}</math>

: <math>\int_0^\infty {e^{-ax^{2}}}\, dx=\frac {1}{2} \sqrt{\frac {\pi}{a}}</math>

: <math>\int_0^\infty {e^{-ax^{2}}}\cos bx\, dx=\frac {1}{2} \sqrt{\frac{\pi}{a}}e^{-b^{2}/4a}</math>

: <math>\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 </math>

: <math>\int_{-\infty}^{+\infty} e^{-(ax^{2}+bx+c)}\ dx=\sqrt {\frac{\pi}{a}}e^{(b^{2}-4ac)/4a}</math>

: <math>\int_0^\infty x^{n}e^{-ax}\ dx=\frac{\Gamma (n+1)}{a^{n+1}}</math>

: <math>\int_0^\infty x^{m}e^{-ax^2}\ dx=\frac{\Gamma [(m+1)/2]}{2a^{(m+1)/2}}</math>

: <math>\int_0^\infty e^{-ax^{2}-b/x^{2}}\ dx=\frac{1}{2} \sqrt \frac{\pi}{a}e^{-2 \sqrt{ab}}</math>

: <math>\int_0^\infty \frac {x}{e^{x}-1}\ dx=\zeta (2)= \frac {\pi^2}{6}</math>

: <math>\int_0^\infty \frac {x^{n-1}}{e^{x}-1}\ dx=\Gamma (n)\zeta (n)</math>

: <math>\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}</math>

: <math>\int_0^\infty \frac {\sin mx}{e^{2\pi x}-1}\ dx=\frac{1}{4} \coth\frac{m}{2}- \frac{1}{2m}</math>

: <math>\int_0^\infty (\frac {1}{1+x}- e^{-x})\frac{dx}{x}=\gamma</math>

: <math>\int_0^\infty \frac {e^{-x^2}-e^{-x}}{x}\ dx=\frac{\gamma}{2}</math>

: <math>\int_0^\infty( \frac {1}{e^x-1}-\frac{e^{-x}}{x})dx=\gamma</math>

: <math>\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}</math>

: <math>\int_0^\infty \frac {e^{-ax}-e^{-bx}}{x \csc px}\ dx=\tan^{-1}\frac{b}{p}-\tan^{-1}\frac{a}{p}</math>

: <math>\int_0^\infty \frac {e^{-ax}(1-\cos x)}{x^2}\ dx=\cot^{-1} a-\frac{a}{2}\ln(a^2+1)</math>

: <math>\int_{-\infty}^\infty e^{-x^2}\,dx=\sqrt{\pi}</math>

: <math>\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 </math>

==Definite integrals involving logarithmic functions==

: <math>\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</math>

: <math>\int_0^1 \frac{\ln x}{1+x}\, dx= -\frac{\pi^2}{12}</math>

: <math>\int_0^1 \frac{\ln x}{1-x}\, dx= -\frac{\pi^2}{6}</math>

: <math>\int_0^1 \frac{\ln (1+x)}{x}\, dx= \frac{\pi^2}{12}</math>

: <math>\int_0^1 \frac{\ln (1-x)}{x}\, dx= -\frac{\pi^2}{6}</math>

==Definite integrals involving hyperbolic functions==

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

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

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

==Miscellaneous definite integrals==

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

<math>\int_{-a}^{a} (a+x)^{m-1}(a-x)^{n-1}\ dx=(2a)^{m+n-1}\frac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}</math>

== See also ==
* [[List of integrals]]
* [[Indefinite sum]]
* [[Gamma function]]
* [[List of limits]]

==References==
{{Reflist|refs
<ref name=schaum>{{cite book|last=Murray R. Spiegel, Seymour Lipschutz, John Liu|title=Mathematical handbook of formulas and tables|year=2009|publisher=McGraw-Hill|isbn=978-0071548557|edition=3rd ed.}}</ref>
<ref name=crc>{{cite book|last=Zwillinger|first=Daniel|title=CRC standard mathematical tables and formulae|year=2003|publisher=CRC Press|isbn=978-1439835487|edition=32nd ed.}}</ref>
<ref name=abram>{{cite book|last=Abramowitz|first=Milton|title=Handbook of mathematical functions with formulas, graphs, and mathematical tables|year=1965|publisher=U.S. Govt. Print. Off.|isbn=978-0486612720|edition=Unabridged and unaltered republ. [der Ausg.] 1964, 5. Dover printing|coauthors=Stegun, Irene A.}}</ref>
}}

[[Category:Integrals|*]]
[[Category:Mathematics-related lists|Integrals]]
[[Category:Mathematical tables|Integrals]]

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Penrose square root law