List of integrals of exponential functions - Revision history

en>Degenerate prodigy: /* Indefinite integral */

2014-12-13T01:02:50Z

Indefinite integral

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	~~The following is a list of~~ [~~[integral~~]~~]s (~~[~~[antiderivative~~]~~] functions)~~ of ~~[[logarithmic function]]s. For~~ a ~~complete list of integral functions, see [[list of integrals]]~~.		More and more people are turning to the diet pill for their weight loss program. This provocative [http://ibaragi-home.info/news/2014/08/20/london-hilton-sedu-hairstyles/ London Hilton Sedu Hairstyles \| Ibaragi Hom] site has endless refreshing suggestions for when to mull over it. After all, why proceed through all the [http://Www.Guardian.Co.uk/search?q=activities activities] of exercising when you"re able to simply take a pill. <br><br>

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	: <~~math~~>~~\int\ln ax\;dx = x\ln ax - x~~<~~/math~~>

	: <~~math~~>~~\int\ln (ax + b)\;dx = \frac{(ax+b)\ln(ax+b) - ax}{a}~~<~~/math~~>

	: <~~math~~>~~\int (\ln x)^2\; dx = x(\ln x)^2 - 2x\ln x + 2x~~<~~/math~~>

	: ~~<math>\int (\ln x)^n\; dx = x\sum^{n}_{k=0}(-1)^{n-k} \frac{n!}{k!}(\ln x)^k<~~/~~math>~~

	~~: <math>\int \frac{dx}{\ln x} = \ln\|\ln x\| + \ln x + \sum^\infty_{k=2}\frac{(\ln x)^k}{k\cdot k!}<~~/~~math>~~

	~~: <math>\int \frac{dx}{(\ln x)^n} = -\frac{x}{(n-1)(\ln x)^{n~~-~~1}} + \frac{1}{n~~-~~1}\int\frac{dx}{(\ln x)^{n~~-~~1}} \qquad\mbox{(for }n\neq 1\mbox{)}<~~/~~math>~~

	~~: <math>\int x^m\ln x\~~;~~dx = x^{m+1}\left(\frac{\ln x}{m+1}-\frac{1}{(m+1)^2}\right) \qquad\mbox{(for }m\neq -1\mbox{)}~~<~~/math~~>

	: <~~math~~>~~\int x^m (\ln x)^n\~~; ~~dx = \frac{x^{m+1}(\ln x)^n}{m+1} - \frac{n}{m+1}\int x^m (\ln x)^{n-1} dx \qquad\mbox{(for }m\neq -1\mbox{)}~~<~~/math~~>

	: <~~math~~>~~\int \frac{(\ln x)^n\; dx}{x} = \frac{(\ln x)^{n+1}}{n+1} \qquad\mbox{(for }n\neq -1\mbox{)}~~<~~/math~~>

	: <~~math~~>~~\int \frac{\ln{x^n}\~~;~~dx}{x} = \frac{(\ln{x^n})^2}{2n} \qquad\mbox{(for }n\neq 0\mbox{)}~~ <~~/math~~>

	: <~~math~~>~~\int \frac{\ln x\~~,~~dx}{x^m} =~~ -~~\frac{\ln x}{(m-1)x^{m-1}}-\frac{1}{(m-1)^2 x^{m-1}} \qquad\mbox{(for }m\neq 1\mbox{)}~~<~~/math~~>

	: <~~math~~>~~\int \frac{(\ln x)^n\; dx}{x^m} = -\frac{(\ln x)^n}{(m-1)x^{m-1}} + \frac{n}{m-1}\int\frac{(\ln x)^{n-1} dx}{x^m} \qquad\mbox{(for }m\neq 1\mbox{)}~~<~~/math~~>

	: <~~math~~>~~\int \frac{x^m\~~; ~~dx}{(\ln x)^n} =~~ -~~\frac{x^{m+1}}{(n-1)(\ln x)^{n-1}} + \frac{m+1}{n-1}\int\frac{x^m dx}{(\ln x)^{n-1}} \qquad\mbox{(for }n\neq 1\mbox{)}~~<~~/math~~>

	: <~~math~~>~~\int \frac{dx}{x\ln x} = \ln \left\|\ln x\right\|~~<~~/math~~>

	: <~~math~~>~~\int \frac{dx}{x^n\ln x} = \ln \left\|\ln x\right\| + \sum^\infty_{k=1} (-1)^k\frac{(n-1)^k(\ln x)^k}{k\cdot k!}~~<~~/math~~>

	: <~~math~~>~~\int \frac{dx}{x(\ln x)^n} =~~ -~~\frac{1}{(n-1)(\ln x)^{n-1}} \qquad\mbox{(for }n\neq 1\mbox{)}~~<~~/math~~>

	: <~~math~~>~~\int \ln(x^2+a^2)\; dx = x\ln(x^2+a^2)-2x+2a\tan^{-1} \frac{x}{a}~~<~~/math~~>

	: <~~math~~>~~\int \frac{x}{x^2+a^2}\ln(x^2+a^2)\~~; ~~dx = \frac{1}{4} \ln^2(x^2+~~a~~^2)~~<~~/math~~>

	: <~~math~~>~~\int \sin (\ln x)\;dx = \frac{x}{2}(\sin (\ln x) - \cos (\ln x))<~~/~~math>~~

	~~: <math>\int \cos (\ln x)\;dx = \frac{x}{2}(\sin (\ln x) + \cos (\ln x))<~~/~~math>~~

	~~: <math>\int e^x \left(x \ln x~~ - x - ~~\frac{1}{x}\right)\;dx = e^x (x \ln x~~ - x - ~~\ln x) <~~/~~math>~~

	: <~~math~~>~~\int \frac{1}{e^x} \left( \frac{1}{x}-\ln x \right)\;dx = \frac{\ln x}{e^x}~~ <~~/math~~>

	: <~~math~~>~~\int e^x \left( \frac{1}{\ln x}- \frac{1}{x\ln^2 x} \right)\;dx = \frac{e^x}{\ln x}~~ <~~/math~~>

	~~For~~ <~~math~~>n<~~/math~~> ~~consecutive integrations~~, the ~~formula~~

	~~: <math>\int\ln x\;dx = x\;(\ln x - 1) +C_{0} </math>~~

	~~generalizes~~ to

	: <~~math~~>~~\int\cdot\cdot\cdot\int\ln x\;dx\cdot\cdot\cdot\;dx = \frac{x^{n}}{n!}\left(\ln\,x-\sum_{k=1}^{n}\frac{1}{k}\right)+ \sum_{k=0}^{n-1} C_{k} \frac{x^{k}}{k!}~~ <~~/math~~>

	~~== References ==~~
	* [[Milton Abramowitz]] and ~~[[Irene A. Stegun]], ''[[Abramowitz and Stegun\|Handbook of Mathematical Functions with Formulas, Graphs~~, and ~~Mathematical Tables]]'', 1964. A few integrals are listed on~~ [http://www.~~math.sfu~~.ca/~~~cbm/aands/page_69~~.~~htm page 69]~~.

	~~{{Lists~~ of ~~integrals}}~~

	[~~[Category~~:~~Integrals\|Logarithmic functions]]~~
	~~[[Category:Mathematics-related lists\|Integrals of logarithmic functions]~~]

149.130.246.51: Undid revision 582340425 by 203.110.246.22 (talk) Because this revision removed an entire section of integrals that was very useful. Not sure why anyone would do that.

2013-11-21T22:23:18Z

Undid revision 582340425 by 203.110.246.22 (talk) Because this revision removed an entire section of integrals that was very useful. Not sure why anyone would do that.

← Older revision		Revision as of 00:23, 22 November 2013
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	~~Jerrie~~ is ~~what you can call me but Simply put i don~~'~~t like when somebody use my full company. My wife~~ and ~~I chose to reside in Massachusetts~~. ~~What I love doing is always to play croquet~~ ~~now I have time period to take on new things. The job I've been occupying suitable~~ for ~~years is an sequence clerk. See what's new on~~ a ~~few website here~~: ~~http~~://~~prometeu.net~~<br><br>~~Visit my homepage~~ - [http://~~prometeu~~.~~net clash~~ of ~~clans cheats ipad gems~~]		The following is a list of [[integral]]s ([[antiderivative]] functions) of [[logarithmic function]]s. For a complete list of integral functions, see [[list of integrals]].

			''Note:'' ''x''>0 is assumed throughout this article, and the [[constant of integration]] is omitted for simplicity.

			: <math>\int\ln ax\;dx = x\ln ax - x</math>

			: <math>\int\ln (ax + b)\;dx = \frac{(ax+b)\ln(ax+b) - ax}{a}</math>

			: <math>\int (\ln x)^2\; dx = x(\ln x)^2 - 2x\ln x + 2x</math>

			: <math>\int (\ln x)^n\; dx = x\sum^{n}_{k=0}(-1)^{n-k} \frac{n!}{k!}(\ln x)^k</math>

			: <math>\int \frac{dx}{\ln x} = \ln\|\ln x\| + \ln x + \sum^\infty_{k=2}\frac{(\ln x)^k}{k\cdot k!}</math>

			: <math>\int \frac{dx}{(\ln x)^n} = -\frac{x}{(n-1)(\ln x)^{n-1}} + \frac{1}{n-1}\int\frac{dx}{(\ln x)^{n-1}} \qquad\mbox{(for }n\neq 1\mbox{)}</math>

			: <math>\int x^m\ln x\;dx = x^{m+1}\left(\frac{\ln x}{m+1}-\frac{1}{(m+1)^2}\right) \qquad\mbox{(for }m\neq -1\mbox{)}</math>

			: <math>\int x^m (\ln x)^n\; dx = \frac{x^{m+1}(\ln x)^n}{m+1} - \frac{n}{m+1}\int x^m (\ln x)^{n-1} dx \qquad\mbox{(for }m\neq -1\mbox{)}</math>

			: <math>\int \frac{(\ln x)^n\; dx}{x} = \frac{(\ln x)^{n+1}}{n+1} \qquad\mbox{(for }n\neq -1\mbox{)}</math>

			: <math>\int \frac{\ln{x^n}\;dx}{x} = \frac{(\ln{x^n})^2}{2n} \qquad\mbox{(for }n\neq 0\mbox{)} </math>

			: <math>\int \frac{\ln x\,dx}{x^m} = -\frac{\ln x}{(m-1)x^{m-1}}-\frac{1}{(m-1)^2 x^{m-1}} \qquad\mbox{(for }m\neq 1\mbox{)}</math>

			: <math>\int \frac{(\ln x)^n\; dx}{x^m} = -\frac{(\ln x)^n}{(m-1)x^{m-1}} + \frac{n}{m-1}\int\frac{(\ln x)^{n-1} dx}{x^m} \qquad\mbox{(for }m\neq 1\mbox{)}</math>

			: <math>\int \frac{x^m\; dx}{(\ln x)^n} = -\frac{x^{m+1}}{(n-1)(\ln x)^{n-1}} + \frac{m+1}{n-1}\int\frac{x^m dx}{(\ln x)^{n-1}} \qquad\mbox{(for }n\neq 1\mbox{)}</math>

			: <math>\int \frac{dx}{x\ln x} = \ln \left\|\ln x\right\|</math>

			: <math>\int \frac{dx}{x^n\ln x} = \ln \left\|\ln x\right\| + \sum^\infty_{k=1} (-1)^k\frac{(n-1)^k(\ln x)^k}{k\cdot k!}</math>

			: <math>\int \frac{dx}{x(\ln x)^n} = -\frac{1}{(n-1)(\ln x)^{n-1}} \qquad\mbox{(for }n\neq 1\mbox{)}</math>

			: <math>\int \ln(x^2+a^2)\; dx = x\ln(x^2+a^2)-2x+2a\tan^{-1} \frac{x}{a}</math>

			: <math>\int \frac{x}{x^2+a^2}\ln(x^2+a^2)\; dx = \frac{1}{4} \ln^2(x^2+a^2)</math>

			: <math>\int \sin (\ln x)\;dx = \frac{x}{2}(\sin (\ln x) - \cos (\ln x))</math>

			: <math>\int \cos (\ln x)\;dx = \frac{x}{2}(\sin (\ln x) + \cos (\ln x))</math>

			: <math>\int e^x \left(x \ln x - x - \frac{1}{x}\right)\;dx = e^x (x \ln x - x - \ln x) </math>

			: <math>\int \frac{1}{e^x} \left( \frac{1}{x}-\ln x \right)\;dx = \frac{\ln x}{e^x} </math>

			: <math>\int e^x \left( \frac{1}{\ln x}- \frac{1}{x\ln^2 x} \right)\;dx = \frac{e^x}{\ln x} </math>

			For <math>n</math> consecutive integrations, the formula

			: <math>\int\ln x\;dx = x\;(\ln x - 1) +C_{0} </math>

			generalizes to

			: <math>\int\cdot\cdot\cdot\int\ln x\;dx\cdot\cdot\cdot\;dx = \frac{x^{n}}{n!}\left(\ln\,x-\sum_{k=1}^{n}\frac{1}{k}\right)+ \sum_{k=0}^{n-1} C_{k} \frac{x^{k}}{k!} </math>

			== References ==
			* [[Milton Abramowitz]] and [[Irene A. Stegun]], ''[[Abramowitz and Stegun\|Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables]]'', 1964. A few integrals are listed on [http://www.math.sfu.ca/~cbm/aands/page_69.htm page 69].

			{{Lists of integrals}}

			[[Category:Integrals\|Logarithmic functions]]
			[[Category:Mathematics-related lists\|Integrals of logarithmic functions]]

en>Blah314: /* Indefinite integrals */

2012-08-01T15:18:04Z

Indefinite integrals

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