Całki funkcji arcus

Lista całek nieoznaczonych arcus

Lista całek nieoznaczonych arcus

Lista innych całek znajduje się w tablicy całek.

Całki funkcji

wymiernych | logarytmicznych | wykładniczych | niewymiernych | trygonometrycznych | hiperbolicznych | arcus | area

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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