Lista de integrales de funciones trigonométricas inversas

Arcoseno

\int \arcsin x\,dx=x\arcsin x+{\sqrt {1-x^{2))}+C

\int \arcsin {\frac {x}{a}}\,dx=x\arcsin {\frac {x}{a}}+{\sqrt {a^{2}-x^{2} }}+C

\int x\arcsin {\frac {x}{a}}\,dx=\left({\frac {x^{2}}{2}}-{\frac {a^{2}} {4}}\right)\arcsen {\frac {x}{a}}+{\frac {x}{4}}{\sqrt {a^{2}-x^{2}}}+C

\int x^{2}\arcsin {\frac {x}{a}}\,dx={\frac {x^{3}}{3}}\arcsin {\frac {x}{a ))+{\frac{x^{2}+2a^{2}}{9}}{\sqrt {a^{2}-x^{2}}}+C

\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}\arcsen x}{n-1}}+n\int x^{n-2}\arcsen x\,dx\ Correcto)

\int \cos ^{n}x\arcsen x\,dx=\left(x^{n^{2}+1}\arccos x+{\frac {x^{n}{\sqrt {1 -x^{4}}}-nx^{n^{2}-1}\arcos x}{n^{2}-1}}+n\int x^{n^{2}-2}\ arccos x\,dx\right)

\int \arccos x\,dx=x\arccos x-{\sqrt {1-x^{2))}+C

\int \arccos {\frac {x}{a}}\,dx=x\arccos {\frac {x}{a}}-{\sqrt {a^{2}-x^{2} }}+C

\int x\arccos {\frac {x}{a}}\,dx=\left({\frac {x^{2}}{2}}-{\frac {a^{2}} {4}}\right)\arcos {\frac {x}{a}}-{\frac {x}{4}}{\sqrt {a^{2}-x^{2}}}+C

\int x^{2}\arcos {\frac {x}{a}}\,dx={\frac {x^{3}}{3}}\arcos {\frac {x}{a }}-{\frac{x^{2}+2a^{2}}{9}}{\sqrt {a^{2}-x^{2}}}+C

\int \operatorname {arctg} \,x\,dx=x\,\operatorname {arctg} \,x-{\frac {1}{2}}\ln(1+x^{2}) +C

\int \operatorname {arctg} \,{\frac {x}{a}}\,dx=x\,\operatorname {arctg} \,{\frac {x}{a}}-{\frac {a}{2}}\ln(1+{\frac{x^{2}}{a^{2}}})+C

\int x\,\operatorname {arctg} \,{\frac {x}{a}}\,dx={\frac {(a^{2}+x^{2})\,\operatorname {arctg} \,{\frac{x}{a}}-ax}{2}}+C

\int x^{2}\,\operatorname {arctg} \,{\frac {x}{a}}\,dx={\frac {x^{3}}{3}}\,\ nombre del operador {arctg} \,{\frac {x}{a}}-{\frac {ax^{2}}{6}}+{\frac {a^{3}}{6}}\ln({ a^{2}+x^{2}})+C

\int x^{n}\,\operatorname {arctg} \,{\frac {x}{a}}\,dx={\frac {x^{n+1}}{n+1} }\,\nombre del operador {arctg} \,{\frac {x}{a}}-{\frac {a}{n+1}}\int {\frac {x^{n+1}}{a^ {2}+x^{2}}}\,dx,\quad n\neq -1

\int \operatorname {arcctg} \,x\,dx=x\,\operatorname {arcctg} \,x+{\frac {1}{2}}\ln(1+x^{2})+ C

\int \operatorname {arcctg} \,{\frac {x}{a}}\,dx=x\,\operatorname {arcctg} \,{\frac {x}{a}}+{\frac {a}{2}}\ln(a^{2}+x^{2})+C

\int x\,\operatorname {arcctg} \,{\frac {x}{a}}\,dx={\frac {a^{2}+x^{2}}{2}}\ ,\nombre del operador {arcctg} \,{\frac {x}{a}}+{\frac {ax}{2}}+C

\int x^{2}\,\operatorname {arcctg} \,{\frac {x}{a}}\,dx={\frac {x^{3}}{3}}\,\ nombre del operador {arcctg} \,{\frac {x}{a}}+{\frac {ax^{2}}{6}}-{\frac {a^{3}}{6}}\ln(a ^{2}+x^{2})+C

\int x^{n}\,\operatorname {arcctg} \,{\frac {x}{a}}\,dx={\frac {x^{n+1}}{n+1} }\,\nombre del operador {arcctg} \,{\frac {x}{a}}+{\frac {a}{n+1}}\int {\frac {x^{n+1}}{a^ {2}+x^{2}}}\,dx,\quad n\neq -1

\int \operatorname {arcsec} x\,dx=x\operatorname {arcsec} x-\ln \left|x+x{\sqrt {{x^{2}-1} \over x^{2 }}}\derecho|+C

\int \operatorname {arcsec} {\frac {x}{a}}\,dx=x\operatorname {arcsec} {\frac {x}{a}}+{\frac {x}{a| x|}}\ln \izquierda|x\pm {\sqrt {x^{2}-1}}\derecha|+C

\int x\operatorname {arcsec} x\,dx={\frac {1}{2}}\left(x^{2}\operatorname {arcsec} x-{\sqrt {x^{2} -1}}\derecha)+C

\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}\nombre del operador {arcsec} x+(1-n)\int x^{n-2}\operatorname {arcsec} x\,dx\right)\right]\right)

\int \operatorname {arccosec} \,x\,dx=x\,\operatorname {arccosec} \,x+\ln \left|x+x{\sqrt {{x^{2}-1} \ sobre x^{2}}}\right|+C

\int \,\operatorname {arccosec} \,{\frac {x}{a}}\,dx=x\,\operatorname {arccosec} \,{\frac {x}{a}}+{ a}\ln {\left({\frac {x}{a}}\left({\sqrt {1-{\frac {a^{2}}{x^{2}}}}}+1\ derecha)\derecha)}+C

\int x\,\operatorname {arccosec} \,{\frac {x}{a}}\,dx={\frac {x^{2}}{2}}\,\operatorname {arccosec} \,{\frac {x}{a}}+{\frac {ax}{2}}{\sqrt {1-{\frac {a^{2}}{x^{2}}}}}+ C

Libros

Gradshtein I. S., Ryzhik I. M. Tablas de integrales, sumas, series y productos. - 4ª ed. - M.: Nauka, 1963. - ISBN 0-12-294757-6 // EqWorld
Dvait G. B. Tablas de integrales San Petersburgo: Editorial e imprenta de JSC VNIIG im. B. V. Vedeneeva, 1995. - 176 p. — ISBN 5-85529-029-8 . // EqMundo
D. Zwillinger. Tablas y fórmulas matemáticas estándar de CRC , 31.ª ed., 2002. ISBN 1-58488-291-3 .
M. Abramowitz y IA Stegun, eds. Manual de funciones matemáticas con fórmulas, gráficos y tablas matemáticas , 1964. ISBN 0-486-61272-4
Korn G. A., Korn T. M. Manual de matemáticas para científicos e ingenieros . - M .: " Nauka ", 1974.

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