Тригонометрилле кутăнла функцисен интегралĕсен йышĕ

Тригонометрилле кутăнла функцисен интегралĕсен йышĕ. Аяларах çавнашкал интегралсен (умсăнарсен) йышне илсе кăтартнă.

Арксинус

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

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

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

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

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

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

Арккосинус

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

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

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

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

Арктангенс

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

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

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

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

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

Арккотангенс

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

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

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

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

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

Арксеканс

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

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

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

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

Арккосеканс

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

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

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

Библиографи

Кĕнекесем

• Градштейн И. С. Рыжик И. М. Таблицы интегралов, сумм, рядов и произведений. — 4-е изд. — М.: Наука, 1963. — ISBN 0-12-294757-6 // EqWorld
• Двайт Г. Б. Таблицы интегралов СПб: Издательство и типография АО ВНИИГ им. Б. В. Веденеева, 1995. — 176 с. — ISBN 5-85529-029-8.
• D. Zwillinger. CRC Standard Mathematical Tables and Formulae, 31st ed., 2002. ISBN 1-58488-291-3.
• M. Abramowitz and I. A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 1964. ISBN 0-486-61272-4

Интегралсен таблицисем

• Интегралы на EqWorld
• S.O.S. Mathematics: Tables and Formulas

Интегралсене шутлани

• The Integrator (Wolfram Research)
• Империя Чисел
• Методы вычисления неопределённых интегралов