Benutzer:Augiasstallputzer~dewikibooks/Printtafel5

Integrale exponentieller Funktionen

1. $\int e^{cx}\;dx={\frac {1}{c}}e^{cx}$
   
   
2. $\int a^{cx}\;dx={\frac {1}{c\ln a}}a^{cx}\qquad {\mbox{(}}a>0,{\mbox{ }}a\neq 1{\mbox{)}}$
   
   
3. $\int xe^{cx}\;dx={\frac {e^{cx}}{c^{2}}}(cx-1)$
   
   
4. $\int x^{2}e^{cx}\;dx=e^{cx}\left({\frac {x^{2}}{c}}-{\frac {2x}{c^{2}}}+{\frac {2}{c^{3}}}\right)$
   
   
5. $\int x^{n}e^{cx}\;dx={\frac {1}{c}}x^{n}e^{cx}-{\frac {n}{c}}\int x^{n-1}e^{cx}dx$
   
   
6. $\int {\frac {e^{cx}\;dx}{x}}=\ln |x|+\sum _{i=1}^{\infty }{\frac {(cx)^{i}}{i\cdot i!}}$
   
   
7. $\int {\frac {e^{cx}\;dx}{x^{n}}}={\frac {1}{n-1}}\left(-{\frac {e^{cx}}{x^{n-1}}}+c\int {\frac {e^{cx}}{x^{n-1}}}\,dx\right)\qquad {\mbox{(}}n\neq 1{\mbox{)}}$
   
   
8. $\int e^{cx}\ln x\;dx={\frac {1}{c}}e^{cx}\ln |x|-\operatorname {Ei} \,(cx)$
   
   
9. $\int e^{cx}\sin bx\;dx={\frac {e^{cx}}{c^{2}+b^{2}}}(c\sin bx-b\cos bx)$
   
   
10. $\int e^{cx}\cos bx\;dx={\frac {e^{cx}}{c^{2}+b^{2}}}(c\cos bx+b\sin bx)$
    
    
11. $\int e^{cx}\sin ^{n}x\;dx={\frac {e^{cx}\sin ^{n-1}x}{c^{2}+n^{2}}}(c\sin x-n\cos x)+{\frac {n(n-1)}{c^{2}+n^{2}}}\int e^{cx}\sin ^{n-2}x\;dx$
    
    
12. $\int e^{cx}\cos ^{n}x\;dx={\frac {e^{cx}\cos ^{n-1}x}{c^{2}+n^{2}}}(c\cos x+n\sin x)+{\frac {n(n-1)}{c^{2}+n^{2}}}\int e^{cx}\cos ^{n-2}x\;dx$
    
    
13. $\int xe^{cx^{2}}\;dx={\frac {1}{2c}}\;e^{cx^{2}}$
    
    
14. $\int x^{n-1}e^{cx^{n}}\;dx={\frac {1}{nc}}\;e^{cx^{n}}\qquad {\mbox{(}}n>1,{\mbox{ }}n\in \mathbb {N} {\mbox{)}}$
15. $\int {1 \over \sigma {\sqrt {2\pi }}}\,e^{-{(x-\mu )^{2}/2\sigma ^{2}}}\;dx={\frac {1}{2\sigma }}(1+{\mbox{erf}}\,{\frac {x-\mu }{\sigma {\sqrt {2}}}})$
    
    
16. $\int e^{x^{2}}\,dx=e^{x^{2}}\left(\sum _{j=0}^{n-1}c_{2j}\,{\frac {1}{x^{2j+1}}}\right)+(2n-1)c_{2n-2}\int {\frac {e^{x^{2}}}{x^{2n}}}\;dx\quad {\mbox{wenn }}n>0,$
    
    wobei $c_{2j}={\frac {1\cdot 3\cdot 5\cdots (2j-1)}{2^{j+1}}}={\frac {2j\,!}{j!\,2^{2j+1}}}\ .$
17. $\int {_{\underbrace {x^{x^{\cdot ^{\cdot ^{x}}}}} } \atop _{m}}dx=\sum _{n=0}^{m}{\frac {(-1)^{n}(n+1)^{n-1}}{n!}}\Gamma (n+1,-\ln x)+\sum _{n=m+1}^{\infty }(-1)^{n}a_{mn}\Gamma (n+1,-\ln x)\qquad {\mbox{(für }}x>0{\mbox{)}}$
    
    wobei $a_{mn}={\begin{cases}1&{\text{wenn }}n=0,\\{\frac {1}{n!}}&{\text{wenn }}m=1,\\{\frac {1}{n}}\sum _{j=1}^{n}ja_{m,n-j}a_{m-1,j-1}&{\text{sonst}}\end{cases}}$
    
    
18. $\int _{-\infty }^{\infty }e^{-ax^{2}}\,dx={\sqrt {\pi \over a}}$ (das Gauß'sche Fehlerintegral)
19. $\int _{-\infty }^{\infty }e^{-ax^{2}}e^{bx}\,dx={\sqrt {\frac {\pi }{a}}}e^{\frac {b^{2}}{4a}}$
    
    
20. $\int _{-\infty }^{\infty }xe^{-a(x-b)^{2}}\,dx=b{\sqrt {\pi \over a}}$
    
    
21. $\int _{-\infty }^{\infty }x^{2}e^{-ax^{2}}\,dx=2\int _{0}^{\infty }x^{2}e^{-ax^{2}}\,dx={\frac {1}{2}}{\sqrt {\frac {\pi }{a^{3}}}}$
    
    
22. $\int _{0}^{\infty }x^{2n}e^{-{x^{2}}/{a^{2}}}\,dx={\sqrt {\pi }}{(2n)! \over {n!}}{\left({\frac {a}{2}}\right)}^{2n+1}$
    
    
23. $\int _{0}^{2\pi }e^{x\cos \vartheta }d\vartheta =2\pi I_{0}(x)$ ($I_{0}$ ist die modifizierte Besselfunktion erster Ordnung)
24. $\int _{0}^{2\pi }e^{x\cos \vartheta +y\sin \vartheta }d\vartheta =2\pi I_{0}\left({\sqrt {x^{2}+y^{2}}}\right)$
    
    
25. $\int _{0}^{\infty }x^{a}e^{-bx}dx={\frac {a!}{b^{a+1}}}$
    
    

Integrale logarithmischer Funktionen

Hinweis

Es wird angenommen, dass x > 0 ist.

101. $\int \ln cx\;dx=x\ln cx-x$
102. $\int \ln(ax+b)\;dx=x\ln(ax+b)-x+{\frac {b}{a}}\ln(ax+b)$
103. $\int (\ln x)^{2}\;dx=x(\ln x)^{2}-2x\ln x+2x$
104. $\int (\ln cx)^{n}\;dx=x(\ln cx)^{n}-n\int (\ln cx)^{n-1}dx$
105. $\int {\frac {dx}{\ln x}}=\ln |\ln x|+\ln x+\sum _{i=2}^{\infty }{\frac {(\ln x)^{i}}{i\cdot i!}}$
106. $\int {\frac {dx}{(\ln x)^{n}}}=-{\frac {x}{(n-1)(\ln x)^{n-1}}}+{\frac {1}{n-1}}\int {\frac {dx}{(\ln x)^{n-1}}}\qquad {\mbox{(}}n\neq 1{\mbox{)}}$
107. $\int x^{m}\ln x\;dx=x^{m+1}\left({\frac {\ln x}{m+1}}-{\frac {1}{(m+1)^{2}}}\right)\qquad {\mbox{(}}m\neq -1{\mbox{)}}$
108. $\int x^{m}(\ln x)^{n}\;dx={\frac {x^{m+1}(\ln x)^{n}}{m+1}}-{\frac {n}{m+1}}\int x^{m}(\ln x)^{n-1}dx\qquad {\mbox{(}}m\neq -1{\mbox{)}}$
109. $\int {\frac {(\ln x)^{n}\;dx}{x}}={\frac {(\ln x)^{n+1}}{n+1}}\qquad {\mbox{(}}n\neq -1{\mbox{)}}$
110. $\int {\frac {\ln x\,dx}{x^{m}}}=-{\frac {\ln x}{(m-1)x^{m-1}}}-{\frac {1}{(m-1)^{2}x^{m-1}}}\qquad {\mbox{(}}m\neq 1{\mbox{)}}$
111. $\int {\frac {(\ln x)^{n}\;dx}{x^{m}}}=-{\frac {(\ln x)^{n}}{(m-1)x^{m-1}}}+{\frac {n}{m-1}}\int {\frac {(\ln x)^{n-1}dx}{x^{m}}}\qquad {\mbox{(}}m\neq 1{\mbox{)}}$
112. $\int {\frac {x^{m}\;dx}{(\ln x)^{n}}}=-{\frac {x^{m+1}}{(n-1)(\ln x)^{n-1}}}+{\frac {m+1}{n-1}}\int {\frac {x^{m}dx}{(\ln x)^{n-1}}}\qquad {\mbox{(}}n\neq 1{\mbox{)}}$
113. $\int {\frac {dx}{x\ln x}}=\ln |\ln x|$
114. $\int {\frac {dx}{x^{n}\ln x}}=\ln |\ln x|+\sum _{i=1}^{\infty }(-1)^{i}{\frac {(n-1)^{i}(\ln x)^{i}}{i\cdot i!}}$
115. $\int {\frac {dx}{x(\ln x)^{n}}}=-{\frac {1}{(n-1)(\ln x)^{n-1}}}\qquad {\mbox{(}}n\neq 1{\mbox{)}}$
116. $\int \sin(\ln x)\;dx={\frac {x}{2}}(\sin(\ln x)-\cos(\ln x))$
117. $\int \cos(\ln x)\;dx={\frac {x}{2}}(\sin(\ln x)+\cos(\ln x))$
118. $\int e^{x}(x\ln x-x-{\frac {1}{x}})\;dx=e^{x}(x\ln x-x-\ln x)$