Тригонометрилле кутăнла функцисен интегралĕсен йышĕ

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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