Danh sách tích phân với hàm lượng giác ngược

Dưới đây là danh sách các tích phân với hàm lượng giác ngược.

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

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

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

$\int x^{n}\sin ^{{-1}}x\,dx={\frac {1}{n+1}}\left(x^{{n+1}}\sin ^{{-1}}x\right.$

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

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

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

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

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

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

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

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

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

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

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

$\left.\left.+(1-n)\left(x^{{n-1}}\operatorname{arcsec} {x}+(1-n)\int x^{{n-2}}\operatorname{arcsec} {x}\,dx\right)\right)\right)$

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

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

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

$\int x^{n}\,{\mathrm {arccot}}\,{\frac {x}{c}}\,dx={\frac {x^{{n+1}}}{n+1}}\,{\mathrm {arccot}}\,{\frac {x}{c}}+{\frac {c}{n+1}}\int {\frac {x^{{n+1}}dx}{c^{2}+x^{2}}}\qquad {\mbox{( }}n\neq 1{\mbox{)}}$

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• Danh sách tích phân

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