Formelsammlung Mathematik: Unbestimmte Integrale der invers-trigonometrischen Funktionen

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Nachfolgende Liste enthält einige Integrale invers-trigonometrischer Funktionen

Anmerkung: Es gibt drei übliche Schreibweisen für invers-trigonometrische Funktionen. Die arcsinus Funktion lässt sich beispielsweise als sin⁻¹, asin, oder wie auf dieser Seite als arcsin darstellen.

Arcussinus

$\int \arcsin {\frac {x}{c}}\ dx=x\arcsin {\frac {x}{c}}+{\sqrt {c^{2}-x^{2}}}$

$\int x\arcsin {\frac {x}{c}}\ dx=\left({\frac {x^{2}}{2}}-{\frac {c^{2}}{4}}\right)\arcsin {\frac {x}{c}}+{\frac {x}{4}}{\sqrt {c^{2}-x^{2}}}$

$\int x^{2}\arcsin {\frac {x}{c}}\ dx={\frac {x^{3}}{3}}\arcsin {\frac {x}{c}}+{\frac {x^{2}+2c^{2}}{9}}{\sqrt {c^{2}-x^{2}}}$

$\int x^{n}\arcsin x\ dx={\frac {1}{n+1}}\left(x^{n+1}\arcsin x+{\frac {x^{n}{\sqrt {1-x^{2}}}-nx^{n-1}\arcsin x}{n-1}}+n\int x^{n-2}\arcsin x\ dx\right)$

Arcuscosinus

$\int \arccos {\frac {x}{c}}\ dx=x\arccos {\frac {x}{c}}-{\sqrt {c^{2}-x^{2}}}$

$\int x\arccos {\frac {x}{c}}\ dx=\left({\frac {x^{2}}{2}}-{\frac {c^{2}}{4}}\right)\arccos {\frac {x}{c}}-{\frac {x}{4}}{\sqrt {c^{2}-x^{2}}}$

$\int x^{2}\arccos {\frac {x}{c}}\ dx={\frac {x^{3}}{3}}\arccos {\frac {x}{c}}-{\frac {x^{2}+2c^{2}}{9}}{\sqrt {c^{2}-x^{2}}}$

Arcustangens

$\int \arctan {\bigg (}{\frac {x}{c}}{\bigg )}dx=x\arctan {\bigg (}{\frac {x}{c}}{\bigg )}-{\frac {c}{2}}\ln(c^{2}+x^{2})$

$\int x\arctan {\bigg (}{\frac {x}{c}}{\bigg )}dx={\frac {(c^{2}+x^{2})\arctan {\bigg (}{\frac {x}{c}}{\bigg )}-cx}{2}}$

$\int x^{2}\arctan {\bigg (}{\frac {x}{c}}{\bigg )}dx={\frac {x^{3}}{3}}\arctan {\bigg (}{\frac {x}{c}}{\bigg )}-{\frac {cx^{2}}{6}}+{\frac {c^{3}}{6}}\ln {c^{2}+x^{2}}$

$\int x^{n}\arctan {\bigg (}{\frac {x}{c}}{\bigg )}dx={\frac {x^{n+1}}{n+1}}\arctan {\bigg (}{\frac {x}{c}}{\bigg )}-{\frac {c}{n+1}}\int {\frac {x^{n+1}}{c^{2}+x^{2}}}\ dx,\quad n\neq 1$

Arcussecans

$\int \operatorname {arcsec} {\frac {x}{c}}\ dx=x\operatorname {arcsec} {\frac {x}{c}}+{\frac {x}{c|x|}}\ln \left|x\pm {\sqrt {x^{2}-1}}\right|$

$\int x\operatorname {arcsec} x\ dx={\frac {1}{2}}\left(x^{2}\operatorname {arcsec} x-{\sqrt {x^{2}-1}}\right)$

$\int x^{n}\operatorname {arcsec} x\ dx={\frac {1}{n+1}}\left(x^{n+1}\operatorname {arcsec} x-{\frac {1}{n}}\left[x^{n-1}{\sqrt {x^{2}-1}}+(1-n)\left(x^{n-1}\operatorname {arcsec} x+(1-n)\int x^{n-2}\operatorname {arcsec} x\ dx\right)\right]\right)$

Arcuscotangens

$\int \operatorname {arccot} {\frac {x}{c}}\ dx=x\operatorname {arccot} {\frac {x}{c}}+{\frac {c}{2}}\ln(c^{2}+x^{2})$

$\int x\operatorname {arccot} {\frac {x}{c}}\ dx={\frac {c^{2}+x^{2}}{2}}\operatorname {arccot} {\frac {x}{c}}+{\frac {cx}{2}}$

$\int x^{2}\operatorname {arccot} {\frac {x}{c}}\ dx={\frac {x^{3}}{3}}\operatorname {arccot} {\frac {x}{c}}+{\frac {cx^{2}}{6}}-{\frac {c^{3}}{6}}\ln(c^{2}+x^{2})$

$\int x^{n}\operatorname {arccot} {\frac {x}{c}}\ dx={\frac {x^{n+1}}{n+1}}\operatorname {arccot} {\frac {x}{c}}+{\frac {c}{n+1}}\int {\frac {x^{n+1}}{c^{2}+x^{2}}}\ dx,\quad n\neq 1$

Arcuscosecans

$\int \operatorname {arccsc} x\ dx=x\operatorname {arccsc} x+\ln {(x({\sqrt {1+{\frac {1}{x^{2}}}}}+1))}$

Siehe auch

• Englische Wikipedia