Wys bron van Lys van integrale van logaritmiese funksies

Hier volg 'n lys van [[integraal|integrale]] ([[anti-afgeleide]] funksies) van [[logaritmiese funksie]]s. Vir 'n volledige lys van integrale funksies, sien [[lys van integrale]].

''Nota:'' ''x'' &gt; 0 word in hierdie artikel deurgaans veronderstel, en die [[konstante van integrasie]] word vir eenvoudigheid weggelaat.

== Integrale wat slegs logaritmiese funksies bevat ==

: <math>\int\log_a x\,dx = \frac{x\ln x - x}{\ln a}</math>

: <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} = \operatorname{li}(x)</math>, die [[logaritmiese integraal]].

: <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{(vir }n\neq 1\mbox{)}</math>

== Integrale wat logaritmiese en magsfunksies bevat ==

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

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

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

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

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

: <math>\int \frac{dx}{x\ln \ln x} = \operatorname{li}(\ln x)</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{(vir }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>

== Integrale wat logaritmiese en trigonometriese funksies bevat ==

: <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>

== Integrale wat logaritmiese en eksponensiële funksies bevat ==

: <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 x)^2} \right)\,dx = \frac{e^x}{\ln x} </math>

== ''n'' opeenvolgende integrasies ==

Vir <math>n</math> opeenvolgende integrasies, veralgemeen die formule

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

na

: <math>\int\dotsi\int\ln x\,dx\dotsm 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>

== Verwysings ==
* [[Milton Abramowitz]] en [[Irene A. Stegun]], ''[[Abramowitz en Stegun|Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables]]'', 1964. 'n Paar integrale word op [http://www.math.sfu.ca/~cbm/aands/page_69.htm bladsy 69] gelys.

{{Lyste van integrale}}

[[Kategorie:Integrale|Logaritmiese funksies]]
[[Kategorie:Lyste|Integrale van logaritmiese funksies]]