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Seznam integrálů inverzních trigonometrických funkcí

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{{Wikipedia-cs|Seznam integrálů inverzních trigonometrických funkcí|700}}
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Toto je seznam [[integrál]]ů (primitivních funkcí) pro integrandy obsahující [[Cyklometrická funkce|inverzní trigonometrické funkce]].
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== [[Arkussinus]] ==
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:<big>\(\int \arcsin \frac{x}{c} \  \mathrm{d}x = x \arcsin \frac{x}{c} + \sqrt{c^2 - x^2}\)</big>
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:<big>\(\int x \arcsin \frac{x}{c} \  \mathrm{d}x = \left( \frac{x^2}{2} - \frac{c^2}{4} \right) \arcsin \frac{x}{c} + \frac{x}{4} \sqrt{c^2 - x^2}\)</big>
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:<big>\(\int x^2 \arcsin \frac{x}{c} \  \mathrm{d}x = \frac{x^3}{3} \arcsin \frac{x}{c} + \frac{x^2 + 2c^2}{9} \sqrt{c^2 - x^2}\)</big>
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:<big>\(\int x^n \arcsin x \  \mathrm{d}x = \frac{1}{n + 1} \left( x^{n + 1} \arcsin x + \frac{x^n \sqrt{1 - x^2} - n x^{n - 1} \arcsin x}{n - 1} + n \int x^{n - 2} \arcsin x \  \mathrm{d}x \right)\)</big>
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== [[Arkuskosinus]] ==
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{{RIGHTTOC}}
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:<big>\(\int \arccos \frac{x}{c} \  \mathrm{d}x = x \arccos \frac{x}{c} - \sqrt{c^2 - x^2}\)</big>
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:<big>\(\int x \arccos \frac{x}{c} \  \mathrm{d}x = \left( \frac{x^2}{2} - \frac{c^2}{4} \right) \arccos \frac{x}{c} - \frac{x}{4} \sqrt{c^2 - x^2}\)</big>
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:<big>\(\int x^2 \arccos \frac{x}{c} \  \mathrm{d}x = \frac{x^3}{3} \arccos \frac{x}{c} - \frac{x^2 + 2c^2}{9} \sqrt{c^2 - x^2}\)</big>
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== [[Arkustangens]] ==
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:<big>\(\int \arctan \frac{x}{c} \  \mathrm{d}x = x \arctan \frac{x}{c} - \frac{c}{2} \ln(c^2 + x^2)\)</big>
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:<big>\(\int x \arctan \frac{x}{c} \  \mathrm{d}x = \frac{ (c^2 + x^2) \arctan \frac{x}{c} - c x}{2}\)</big>
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:<big>\(\int x^2 \arctan \frac{x}{c} \  \mathrm{d}x = \frac{x^3}{3} \arctan \frac{x}{c} - \frac{c x^2}{6} + \frac{c^3}{6} \ln(c^2 + x^2)\)</big>
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:<big>\(\int x^n \arctan \frac{x}{c} \  \mathrm{d}x = \frac{x^{n + 1}}{n + 1} \arctan \frac{x}{c} - \frac{c}{n + 1} \int \frac{x^{n + 1}}{c^2 + x^2} \  \mathrm{d}x, \quad n \neq 1\)</big>
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== [[Arkussekans]] ==
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:<big>\(\int \operatorname {arcsec} \frac{x}{c} \  \mathrm{d}x = x \operatorname {arcsec} \frac{x}{c} + \frac{x}{c |x|} \ln \left| x \pm \sqrt{x^2 - 1} \right|\)</big>
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:<big>\(\int x \operatorname {arcsec} x \  \mathrm{d}x = \frac{1}{2} \left( x^2 \operatorname {arcsec} x - \sqrt{x^2 - 1} \right)\)</big>
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:<big>\(\int x^n \operatorname {arcsec} x \  \mathrm{d}x = \frac{1}{n + 1} \left( x^{n + 1} \operatorname {arcsec} x - \frac{1}{n} \left[ x^{n - 1} \sqrt{x^2 - 1} + (1 - n) \left( x^{n - 1} \operatorname {arcsec} x + (1 - n) \int x^{n - 2} \operatorname {arcsec} x \  \mathrm{d}x \right) \right] \right)\)</big>
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== [[Arkuskotangens]] ==
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:<big>\(\int \operatorname {arccot} \frac{x}{c} \  \mathrm{d}x = x \operatorname {arccot} \frac{x}{c} + \frac{c}{2} \ln(c^2 + x^2)\)</big>
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:<big>\(\int x \operatorname {arccot} \frac{x}{c} \  \mathrm{d}x = \frac{c^2 + x^2}{2} \operatorname {arccot} \frac{x}{c} + \frac{c x}{2}\)</big>
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:<big>\(\int x^2 \operatorname {arccot} \frac{x}{c} \  \mathrm{d}x = \frac{x^3}{3} \operatorname {arccot} \frac{x}{c} + \frac{c x^2}{6} - \frac{c^3}{6} \ln(c^2 + x^2)\)</big>
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:<big>\(\int x^n \operatorname {arccot} \frac{x}{c} \  \mathrm{d}x = \frac{x^{n + 1}}{n+1} \operatorname {arccot} \frac{x}{c} + \frac{c}{n + 1} \int \frac{x^{n + 1}}{c^2 + x^2} \ \mathrm{d}x, \quad n \neq 1\)</big>
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== Externí odkazy ==
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{{Článek z Wikipedie}}
[[Kategorie:Integrální počet]]
[[Kategorie:Integrální počet]]

Aktuální verze z 27. 4. 2025, 10:52

Toto je seznam integrálů (primitivních funkcí) pro integrandy obsahující inverzní trigonometrické funkce.

Arkussinus

\(\int \arcsin \frac{x}{c} \ \mathrm{d}x = x \arcsin \frac{x}{c} + \sqrt{c^2 - x^2}\)
\(\int x \arcsin \frac{x}{c} \ \mathrm{d}x = \left( \frac{x^2}{2} - \frac{c^2}{4} \right) \arcsin \frac{x}{c} + \frac{x}{4} \sqrt{c^2 - x^2}\)
\(\int x^2 \arcsin \frac{x}{c} \ \mathrm{d}x = \frac{x^3}{3} \arcsin \frac{x}{c} + \frac{x^2 + 2c^2}{9} \sqrt{c^2 - x^2}\)
\(\int x^n \arcsin x \ \mathrm{d}x = \frac{1}{n + 1} \left( x^{n + 1} \arcsin x + \frac{x^n \sqrt{1 - x^2} - n x^{n - 1} \arcsin x}{n - 1} + n \int x^{n - 2} \arcsin x \ \mathrm{d}x \right)\)

Arkuskosinus

Obsah

\(\int \arccos \frac{x}{c} \ \mathrm{d}x = x \arccos \frac{x}{c} - \sqrt{c^2 - x^2}\)
\(\int x \arccos \frac{x}{c} \ \mathrm{d}x = \left( \frac{x^2}{2} - \frac{c^2}{4} \right) \arccos \frac{x}{c} - \frac{x}{4} \sqrt{c^2 - x^2}\)
\(\int x^2 \arccos \frac{x}{c} \ \mathrm{d}x = \frac{x^3}{3} \arccos \frac{x}{c} - \frac{x^2 + 2c^2}{9} \sqrt{c^2 - x^2}\)

Arkustangens

\(\int \arctan \frac{x}{c} \ \mathrm{d}x = x \arctan \frac{x}{c} - \frac{c}{2} \ln(c^2 + x^2)\)
\(\int x \arctan \frac{x}{c} \ \mathrm{d}x = \frac{ (c^2 + x^2) \arctan \frac{x}{c} - c x}{2}\)
\(\int x^2 \arctan \frac{x}{c} \ \mathrm{d}x = \frac{x^3}{3} \arctan \frac{x}{c} - \frac{c x^2}{6} + \frac{c^3}{6} \ln(c^2 + x^2)\)
\(\int x^n \arctan \frac{x}{c} \ \mathrm{d}x = \frac{x^{n + 1}}{n + 1} \arctan \frac{x}{c} - \frac{c}{n + 1} \int \frac{x^{n + 1}}{c^2 + x^2} \ \mathrm{d}x, \quad n \neq 1\)

Arkussekans

\(\int \operatorname {arcsec} \frac{x}{c} \ \mathrm{d}x = x \operatorname {arcsec} \frac{x}{c} + \frac{x}{c |x|} \ln \left| x \pm \sqrt{x^2 - 1} \right|\)
\(\int x \operatorname {arcsec} x \ \mathrm{d}x = \frac{1}{2} \left( x^2 \operatorname {arcsec} x - \sqrt{x^2 - 1} \right)\)
\(\int x^n \operatorname {arcsec} x \ \mathrm{d}x = \frac{1}{n + 1} \left( x^{n + 1} \operatorname {arcsec} x - \frac{1}{n} \left[ x^{n - 1} \sqrt{x^2 - 1} + (1 - n) \left( x^{n - 1} \operatorname {arcsec} x + (1 - n) \int x^{n - 2} \operatorname {arcsec} x \ \mathrm{d}x \right) \right] \right)\)

Arkuskotangens

\(\int \operatorname {arccot} \frac{x}{c} \ \mathrm{d}x = x \operatorname {arccot} \frac{x}{c} + \frac{c}{2} \ln(c^2 + x^2)\)
\(\int x \operatorname {arccot} \frac{x}{c} \ \mathrm{d}x = \frac{c^2 + x^2}{2} \operatorname {arccot} \frac{x}{c} + \frac{c x}{2}\)
\(\int x^2 \operatorname {arccot} \frac{x}{c} \ \mathrm{d}x = \frac{x^3}{3} \operatorname {arccot} \frac{x}{c} + \frac{c x^2}{6} - \frac{c^3}{6} \ln(c^2 + x^2)\)
\(\int x^n \operatorname {arccot} \frac{x}{c} \ \mathrm{d}x = \frac{x^{n + 1}}{n+1} \operatorname {arccot} \frac{x}{c} + \frac{c}{n + 1} \int \frac{x^{n + 1}}{c^2 + x^2} \ \mathrm{d}x, \quad n \neq 1\)

Externí odkazy

Citováno z „http://www1fdc.multimediaexpo.cz/mmecz/index.php/Seznam_integr%C3%A1l%C5%AF_inverzn%C3%ADch_trigonometrick%C3%BDch_funkc%C3%AD“