Benutzer:Augiasstallputzer~dewikibooks/Printtafel4

Integrale hyperbolischer Funktionen

1. $\int \sinh cx\,dx={\frac {1}{c}}\cosh cx$
   
   
2. $\int \cosh cx\,dx={\frac {1}{c}}\sinh cx$
   
   
3. $\int \sinh ^{2}cx\,dx={\frac {1}{2c}}\sinh cx\cosh cx-{\frac {x}{2}}$
   
   
4. $\int \cosh ^{2}cx\,dx={\frac {1}{2c}}\sinh cx\cosh cx+{\frac {x}{2}}$
   
   
5. $\int \sinh ^{n}cx\,dx={\frac {1}{cn}}\sinh ^{n-1}cx\cosh cx-{\frac {n-1}{n}}\int \sinh ^{n-2}cx\,dx\qquad {\mbox{( }}n>0{\mbox{)}}$
   
   
6. oder: $\int \sinh ^{n}cx\,dx={\frac {1}{c(n+1)}}\sinh ^{n+1}cx\cosh cx-{\frac {n+2}{n+1}}\int \sinh ^{n+2}cx\,dx\qquad {\mbox{( }}n<0{\mbox{, }}n\neq -1{\mbox{)}}$
   
   
7. $\int \cosh ^{n}cx\,dx={\frac {1}{cn}}\sinh cx\cosh ^{n-1}cx+{\frac {n-1}{n}}\int \cosh ^{n-2}cx\,dx\qquad {\mbox{( }}n>0{\mbox{)}}$
   
   
8. oder: $\int \cosh ^{n}cx\,dx=-{\frac {1}{c(n+1)}}\sinh cx\cosh ^{n+1}cx-{\frac {n+2}{n+1}}\int \cosh ^{n+2}cx\,dx\qquad {\mbox{( }}n<0{\mbox{, }}n\neq -1{\mbox{)}}$
   
   
9. $\int {\frac {dx}{\sinh cx}}={\frac {1}{c}}\ln \left|\tanh {\frac {cx}{2}}\right|$
   
   
10. oder: $\int {\frac {dx}{\sinh cx}}={\frac {1}{c}}\ln \left|{\frac {\cosh cx-1}{\sinh cx}}\right|$
    
    
11. oder: $\int {\frac {dx}{\sinh cx}}={\frac {1}{c}}\ln \left|{\frac {\sinh cx}{\cosh cx+1}}\right|$
    
    
12. oder: $\int {\frac {dx}{\sinh cx}}={\frac {1}{c}}\ln \left|{\frac {\cosh cx-1}{\cosh cx+1}}\right|$
    
    
13. $\int {\frac {dx}{\cosh cx}}={\frac {2}{c}}\arctan e^{cx}$
    
    
14. $\int {\frac {dx}{\sinh ^{n}cx}}={\frac {\cosh cx}{c(n-1)\sinh ^{n-1}cx}}-{\frac {n-2}{n-1}}\int {\frac {dx}{\sinh ^{n-2}cx}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}$
    
    
15. $\int {\frac {dx}{\cosh ^{n}cx}}={\frac {\sinh cx}{c(n-1)\cosh ^{n-1}cx}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\cosh ^{n-2}cx}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}$
    
    
16. $\int {\frac {\cosh ^{n}cx}{\sinh ^{m}cx}}dx={\frac {\cosh ^{n-1}cx}{c(n-m)\sinh ^{m-1}cx}}+{\frac {n-1}{n-m}}\int {\frac {\cosh ^{n-2}cx}{\sinh ^{m}cx}}dx\qquad {\mbox{( }}m\neq n{\mbox{)}}$
    
    
17. oder: $\int {\frac {\cosh ^{n}cx}{\sinh ^{m}cx}}dx=-{\frac {\cosh ^{n+1}cx}{c(m-1)\sinh ^{m-1}cx}}+{\frac {n-m+2}{m-1}}\int {\frac {\cosh ^{n}cx}{\sinh ^{m-2}cx}}dx\qquad {\mbox{( }}m\neq 1{\mbox{)}}$
    
    
18. oder: $\int {\frac {\cosh ^{n}cx}{\sinh ^{m}cx}}dx=-{\frac {\cosh ^{n-1}cx}{c(m-1)\sinh ^{m-1}cx}}+{\frac {n-1}{m-1}}\int {\frac {\cosh ^{n-2}cx}{\sinh ^{m-2}cx}}dx\qquad {\mbox{( }}m\neq 1{\mbox{)}}$
    
    
19. $\int {\frac {\sinh ^{m}cx}{\cosh ^{n}cx}}dx={\frac {\sinh ^{m-1}cx}{c(m-n)\cosh ^{n-1}cx}}+{\frac {m-1}{m-n}}\int {\frac {\sinh ^{m-2}cx}{\cosh ^{n}cx}}dx\qquad {\mbox{( }}m\neq n{\mbox{)}}$
    
    
20. oder: $\int {\frac {\sinh ^{m}cx}{\cosh ^{n}cx}}dx={\frac {\sinh ^{m+1}cx}{c(n-1)\cosh ^{n-1}cx}}+{\frac {m-n+2}{n-1}}\int {\frac {\sinh ^{m}cx}{\cosh ^{n-2}cx}}dx\qquad {\mbox{( }}n\neq 1{\mbox{)}}$
    
    
21. oder: $\int {\frac {\sinh ^{m}cx}{\cosh ^{n}cx}}dx=-{\frac {\sinh ^{m-1}cx}{c(n-1)\cosh ^{n-1}cx}}+{\frac {m-1}{n-1}}\int {\frac {\sinh ^{m-2}cx}{\cosh ^{n-2}cx}}dx\qquad {\mbox{( }}n\neq 1{\mbox{)}}$
    
    
22. $\int x\sinh cx\,dx={\frac {1}{c}}x\cosh cx-{\frac {1}{c^{2}}}\sinh cx$
    
    
23. $\int x\cosh cx\,dx={\frac {1}{c}}x\sinh cx-{\frac {1}{c^{2}}}\cosh cx$
    
    
24. $\int \tanh cx\,dx={\frac {1}{c}}\ln |\cosh cx|$
    
    
25. $\int \coth cx\,dx={\frac {1}{c}}\ln |\sinh cx|$
    
    
26. $\int \tanh ^{n}cx\,dx=-{\frac {1}{c(n-1)}}\tanh ^{n-1}cx+\int \tanh ^{n-2}cx\,dx\qquad {\mbox{( }}n\neq 1{\mbox{)}}$
    
    
27. $\int \coth ^{n}cx\,dx=-{\frac {1}{c(n-1)}}\coth ^{n-1}cx+\int \coth ^{n-2}cx\,dx\qquad {\mbox{( }}n\neq 1{\mbox{)}}$
    
    
28. $\int \sinh bx\sinh cx\,dx={\frac {1}{b^{2}-c^{2}}}(b\sinh cx\cosh bx-c\cosh cx\sinh bx)\qquad {\mbox{( }}b^{2}\neq c^{2}{\mbox{)}}$
    
    
29. $\int \cosh bx\cosh cx\,dx={\frac {1}{b^{2}-c^{2}}}(b\sinh bx\cosh cx-c\sinh cx\cosh bx)\qquad {\mbox{( }}b^{2}\neq c^{2}{\mbox{)}}$
    
    
30. $\int \cosh bx\sinh cx\,dx={\frac {1}{b^{2}-c^{2}}}(b\sinh bx\sinh cx-c\cosh bx\cosh cx)\qquad {\mbox{( }}b^{2}\neq c^{2}{\mbox{)}}$
    
    
31. $\int \sinh(ax+b)\sin(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\cosh(ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2}}}\sinh(ax+b)\cos(cx+d)$
    
    
32. $\int \sinh(ax+b)\cos(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\cosh(ax+b)\cos(cx+d)+{\frac {c}{a^{2}+c^{2}}}\sinh(ax+b)\sin(cx+d)$
    
    
33. $\int \cosh(ax+b)\sin(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\sinh(ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2}}}\cosh(ax+b)\cos(cx+d)$
    
    
34. $\int \cosh(ax+b)\cos(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\sinh(ax+b)\cos(cx+d)+{\frac {c}{a^{2}+c^{2}}}\cosh(ax+b)\sin(cx+d)$
    
    

Integrale von Areafunktionen

101. $\int \mathrm {arsinh} \,{\frac {x}{c}}\,dx=x\,\mathrm {arsinh} \,{\frac {x}{c}}-{\sqrt {x^{2}+c^{2}}}$
     
     
102. $\int \mathrm {arcosh} \,{\frac {x}{c}}\,dx=x\,\mathrm {arcosh} \,{\frac {x}{c}}-{\sqrt {x^{2}-c^{2}}}$
     
     
103. $\int \mathrm {artanh} \,{\frac {x}{c}}\,dx=x\,\mathrm {artanh} \,{\frac {x}{c}}+{\frac {c}{2}}\ln |c^{2}-x^{2}|\qquad {\mbox{(}}|x|<|c|{\mbox{)}}$
     
     
104. $\int \mathrm {arcoth} \,{\frac {x}{c}}\,dx=x\,\mathrm {arcoth} \,{\frac {x}{c}}+{\frac {c}{2}}\ln |x^{2}-c^{2}|\qquad {\mbox{(}}|x|>|c|{\mbox{)}}$
     
     
105. $\int \mathrm {arsech} \,{\frac {x}{c}}\,dx=x\,\mathrm {arsech} \,{\frac {x}{c}}-c\,\mathrm {arctan} \,{\frac {x\,{\sqrt {\frac {c-x}{c+x}}}}{x-c}}\qquad {\mbox{(}}x\in (0,\,c){\mbox{)}}$
     
     
106. $\int \mathrm {arcsch} \,{\frac {x}{c}}\,dx=x\,\mathrm {arcsch} \,{\frac {x}{c}}+c\,\ln \,{\frac {x+{\sqrt {x^{2}+c^{2}}}}{c}}\qquad {\mbox{(}}x\in (0,\,c){\mbox{)}}$