Całki funkcji niewymiernych

Lista całek nieoznaczonych funkcji niewymiernych

Lista całek nieoznaczonych funkcji niewymiernych

Lista innych całek znajduje się w tablicy całek.

Całki funkcji

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

(A01) $\qquad \int {\sqrt {a^{2}-x^{2}}}\;dx={\frac {x}{2}}{\sqrt {a^{2}-x^{2}}}+{\frac {a^{2}}{2}}\arcsin {\frac {x}{|a|}}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}$

(A02) $\int x{\sqrt {a^{2}-x^{2}}}\;dx=-{\frac {1}{3}}{\sqrt {(a^{2}-x^{2})^{3}}}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}$

(A03) $\int x^{2}{\sqrt {a^{2}-x^{2}}}\;dx={\frac {1}{8}}a^{4}\arcsin {\frac {x}{|a|}}+{\frac {1}{8}}x(2x^{2}-a^{2}){\sqrt {a^{2}-x^{2}}}\qquad {\mbox{(}}|x|<|a|{\mbox{)}}$

(A04) $\int {\frac {{\sqrt {a^{2}-x^{2}}}\;dx}{x}}={\sqrt {a^{2}-x^{2}}}-a\ln \left|{\frac {a+{\sqrt {a^{2}-x^{2}}}}{x}}\right|={\sqrt {a^{2}-x^{2}}}+a\ln \left|{\frac {a-{\sqrt {a^{2}-x^{2}}}}{x}}\right|\qquad {\mbox{(}}0<|x|\leq |a|{\mbox{)}}$

(A05) $\int {\frac {{\sqrt {a^{2}-x^{2}}}\;dx}{x^{2}}}=-\arcsin {\frac {x}{|a|}}-{\frac {\sqrt {a^{2}-x^{2}}}{x}}\qquad {\mbox{(}}0<|x|\leq |a|{\mbox{)}}$

(A06) $\int {\frac {dx}{\sqrt {a^{2}-x^{2}}}}=\arcsin {\frac {x}{|a|}}\qquad {\mbox{(}}|x|<|a|{\mbox{)}}$

(A07) $\int {\frac {x\;dx}{\sqrt {a^{2}-x^{2}}}}=-{\sqrt {a^{2}-x^{2}}}\qquad {\mbox{(}}|x|<|a|{\mbox{)}}$

(A08) $\int {\frac {x^{2}\;dx}{\sqrt {a^{2}-x^{2}}}}=-{\frac {x}{2}}{\sqrt {a^{2}-x^{2}}}+{\frac {a^{2}}{2}}\arcsin {\frac {x}{|a|}}\qquad {\mbox{(}}|x|<|a|{\mbox{)}}$

(A09) $\int {\frac {dx}{x{\sqrt {a^{2}-x^{2}}}}}=-{\frac {1}{a}}\ln \left|{\frac {a+{\sqrt {a^{2}-x^{2}}}}{x}}\right|={\frac {1}{a}}\ln \left|{\frac {a-{\sqrt {a^{2}-x^{2}}}}{x}}\right|\qquad {\mbox{(}}0<|x|<|a|{\mbox{)}}$

(A10) $\int {\frac {dx}{x^{2}{\sqrt {a^{2}-x^{2}}}}}=-{\frac {\sqrt {a^{2}-x^{2}}}{a^{2}x}}\qquad {\mbox{(}}0<|x|<|a|{\mbox{)}}$

(B01) $\int {\sqrt {x^{2}+a^{2}}}\;dx={\frac {x}{2}}{\sqrt {x^{2}+a^{2}}}+{\frac {a^{2}}{2}}\,\ln \left(x+{\sqrt {x^{2}+a^{2}}}\right)={\frac {x}{2}}{\sqrt {x^{2}+a^{2}}}-{\frac {a^{2}}{2}}\,\ln \left({\sqrt {x^{2}+a^{2}}}-x\right)$

(B01a)$\int {\sqrt {x^{2}+a^{2}}}\;dx={\frac {x}{2}}{\sqrt {x^{2}+a^{2}}}+{\frac {a^{2}}{2}}\,\mathrm {arsinh} {\frac {x}{|a|}}$

(B02) $\int x{\sqrt {x^{2}+a^{2}}}\;dx={\frac {1}{3}}{\sqrt {(x^{2}+a^{2})^{3}}}$

(B03) $\int x^{2}{\sqrt {x^{2}+a^{2}}}\;dx={\frac {1}{8}}\;x(2x^{2}+a^{2}){\sqrt {x^{2}+a^{2}}}-{\frac {a^{4}}{8}}\ln \left(x+{\sqrt {x^{2}+a^{2}}}\right)$

(B04) $\int {\frac {{\sqrt {x^{2}+a^{2}}}\;dx}{x}}={\sqrt {x^{2}+a^{2}}}-a\ln \left|{\frac {a+{\sqrt {x^{2}+a^{2}}}}{x}}\right|={\sqrt {x^{2}+a^{2}}}+a\ln \left|{\frac {{\sqrt {x^{2}+a^{2}}}-a}{x}}\right|$

(B05) $\int {\frac {{\sqrt {x^{2}+a^{2}}}\;dx}{x^{2}}}=\ln \left(x+{\sqrt {x^{2}+a^{2}}}\right)-{\frac {\sqrt {x^{2}+a^{2}}}{x}}=-\ln \left({\sqrt {x^{2}+a^{2}}}-x\right)-{\frac {\sqrt {x^{2}+a^{2}}}{x}}$

(B06) $\int {\frac {dx}{\sqrt {x^{2}+a^{2}}}}=\ln \left(x+{\sqrt {x^{2}+a^{2}}}\right)=-\ln \left({\sqrt {x^{2}+a^{2}}}-x\right)=\mathrm {arsinh} {\frac {x}{|a|}}$

(B07) $\int {\frac {x\,dx}{\sqrt {x^{2}+a^{2}}}}={\sqrt {x^{2}+a^{2}}}$

(B08) $\int {\frac {x^{2}\;dx}{\sqrt {x^{2}+a^{2}}}}={\frac {x}{2}}{\sqrt {x^{2}+a^{2}}}-{\frac {a^{2}}{2}}\ln \left(x+{\sqrt {x^{2}+a^{2}}}\right)={\frac {x}{2}}{\sqrt {x^{2}+a^{2}}}+{\frac {a^{2}}{2}}\ln \left({\sqrt {x^{2}+a^{2}}}-x\right)={\frac {x}{2}}{\sqrt {x^{2}+a^{2}}}-{\frac {a^{2}}{2}}\,\mathrm {arsinh} {\frac {x}{|a|}}$

(B09) $\int {\frac {dx}{x{\sqrt {x^{2}+a^{2}}}}}=-{\frac {1}{a}}\,\mathrm {arsinh} {\frac {a}{x}}=-{\frac {1}{a}}\ln \left|{\frac {a+{\sqrt {x^{2}+a^{2}}}}{x}}\right|={\frac {1}{a}}\ln \left|{\frac {{\sqrt {x^{2}+a^{2}}}-a}{x}}\right|$

(B10) $\int {\frac {dx}{x^{2}{\sqrt {x^{2}+a^{2}}}}}=-{\frac {\sqrt {x^{2}+a^{2}}}{a^{2}x}}$

(C01) $\int {\sqrt {x^{2}-a^{2}}}\;dx={\frac {x}{2}}{\sqrt {x^{2}-a^{2}}}-{\frac {a^{2}}{2}}\ln \left|x+{\sqrt {x^{2}-a^{2}}}\right|={\frac {x}{2}}{\sqrt {x^{2}-a^{2}}}+{\frac {a^{2}}{2}}\ln \left|x-{\sqrt {x^{2}-a^{2}}}\right|={\frac {x}{2}}{\sqrt {x^{2}-a^{2}}}-{\frac {a^{2}}{2}}\operatorname {sgn} x\,\mathrm {arcosh} \left|{\frac {x}{a}}\right|\qquad {\mbox{(dla }}|x|\geq |a|{\mbox{)}}$

(C02) $\int x{\sqrt {x^{2}-a^{2}}}\;dx={\frac {1}{3}}{\sqrt {(x^{2}-a^{2})^{3}}}\qquad {\mbox{(dla }}|x|\geq |a|{\mbox{)}}$

(C03) $\int x^{2}{\sqrt {x^{2}-a^{2}}}\;dx={\frac {1}{8}}x(2x^{2}-a^{2}){\sqrt {x^{2}-a^{2}}}-{\frac {1}{8}}a^{4}\ln \left|x+{\sqrt {x^{2}-a^{2}}}\right|\qquad {\mbox{(dla }}|x|\geq |a|{\mbox{)}}$

(C04) $\int {\frac {{\sqrt {x^{2}-a^{2}}}\;dx}{x}}={\sqrt {x^{2}-a^{2}}}+a\arcsin {\frac {a}{|x|}}={\sqrt {x^{2}-a^{2}}}-a\arccos {\frac {a}{|x|}}={\sqrt {x^{2}-a^{2}}}-a\operatorname {arctg} {\frac {\sqrt {x^{2}-a^{2}}}{a}}\qquad {\mbox{(dla }}|x|\geq |a|{\mbox{)}}$

(C05) $\int {\frac {{\sqrt {x^{2}-a^{2}}}\;dx}{x^{2}}}=\ln \left|x+{\sqrt {x^{2}-a^{2}}}\right|-{\frac {\sqrt {x^{2}-a^{2}}}{x}}=-\ln \left|x-{\sqrt {x^{2}-a^{2}}}\right|-{\frac {\sqrt {x^{2}-a^{2}}}{x}}\qquad {\mbox{(dla }}|x|\geq |a|{\mbox{)}}$

(C06) $\int {\frac {dx}{\sqrt {x^{2}-a^{2}}}}=\ln \left|x+{\sqrt {x^{2}-a^{2}}}\right|=-\ln \left|x-{\sqrt {x^{2}-a^{2}}}\right|=\operatorname {sgn} x\mathrm {arcosh} \left|{\frac {x}{a}}\right|\qquad {\mbox{(dla }}|x|>|a|{\mbox{)}}$

(C07) $\int {\frac {x\;dx}{\sqrt {x^{2}-a^{2}}}}={\sqrt {x^{2}-a^{2}}}\qquad {\mbox{(dla }}|x|>|a|{\mbox{)}}$

(C08) $\int {\frac {x^{2}\,dx}{\sqrt {x^{2}-a^{2}}}}={\frac {x}{2}}{\sqrt {x^{2}-a^{2}}}+{\frac {a^{2}}{2}}\ln \left|x+{\sqrt {x^{2}-a^{2}}}\right|={\frac {x}{2}}{\sqrt {x^{2}-a^{2}}}-{\frac {a^{2}}{2}}\ln \left|x-{\sqrt {x^{2}-a^{2}}}\right|={\frac {x}{2}}{\sqrt {x^{2}-a^{2}}}+{\frac {a^{2}}{2}}\,\operatorname {sgn} x\,\mathrm {arcosh} \left|{\frac {x}{a}}\right|\qquad {\mbox{(dla }}|x|>|a|{\mbox{)}}$

(C09) $\int {\frac {dx}{x{\sqrt {x^{2}-a^{2}}}}}=-{\frac {1}{a}}\,\arcsin {\frac {a}{|x|}}={\frac {1}{a}}\,\arccos {\frac {a}{|x|}}={\frac {1}{a}}\,\operatorname {arctg} {\frac {\sqrt {x^{2}-a^{2}}}{a}}\qquad {\mbox{(dla }}|x|>|a|{\mbox{)}}$

(C10) $\int {\frac {dx}{x^{2}{\sqrt {x^{2}-a^{2}}}}}={\frac {\sqrt {x^{2}-a^{2}}}{a^{2}x}}\qquad {\mbox{(dla }}|x|>|a|{\mbox{)}}$

Następujące wzory są rozwinięciem wzorów (A06), (B06), (C06), dlatego została im nadana numeracja typu (D06x):

(D06a) $\int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}={\frac {1}{\sqrt {a}}}\ln \left|2{\sqrt {a(ax^{2}+bx+c)}}+2ax+b\right|\qquad {\mbox{(dla }}a>0{\mbox{)}}$

(D06b) $\int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}={\frac {1}{\sqrt {a}}}\,\mathrm {arsinh} {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}\qquad {\mbox{(dla }}a>0{\mbox{, }}4ac-b^{2}>0{\mbox{)}}$

(D06c) $\int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}={\frac {1}{\sqrt {a}}}\ln |2ax+b|\qquad {\mbox{(dla }}a>0{\mbox{, }}4ac-b^{2}=0{\mbox{)}}$

(D06d) $\int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}=-{\frac {1}{\sqrt {-a}}}\arcsin {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}\qquad {\mbox{(dla }}a<0{\mbox{, }}4ac-b^{2}<0{\mbox{)}}$

(D06e) $\int {\frac {x\;dx}{\sqrt {ax^{2}+bx+c}}}={\frac {\sqrt {ax^{2}+bx+c}}{a}}-{\frac {b}{2a}}\int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}$

Uwaga. Wyrażenia podane jako różne wyniki dla danej całki niekoniecznie muszą być równe - mogą się różnić o pewną stałą (przy czym w poszczególnych przedziałach stałe te mogą być różne).

(E01) $\int {\sqrt[{n}]{ax+b}}\;dx={\frac {n}{a\left(n+1\right)}}{\sqrt[{n}]{\left(ax+b\right)^{n+1}}}+C\qquad {\mbox{(dla }}n\geq 2{\mbox{)}}$

(E02) $\int x{\sqrt[{n}]{ax+b}}\;dx={\frac {n}{a^{2}\left(2n+1\right)}}{\sqrt[{n}]{\left(ax+b\right)^{2n+1}}}-{\frac {nb}{a^{2}\left(n+1\right)}}{\sqrt[{n}]{\left(ax+b\right)^{n+1}}}+C\qquad {\mbox{(dla }}n\geq 2{\mbox{)}}$

(E03) $\int x^{2}{\sqrt[{n}]{ax+b}}\;dx={\frac {n}{a^{3}\left(3n+1\right)}}{\sqrt[{n}]{\left(ax+b\right)^{3n+1}}}-{\frac {2nb}{a^{3}\left(2n+1\right)}}{\sqrt[{n}]{\left(ax+b\right)^{2n+1}}}+{\frac {nb^{2}}{a^{3}\left(n+1\right)}}{\sqrt[{n}]{\left(ax+b\right)^{n+1}}}+C\qquad {\mbox{(dla }}n\geq 2{\mbox{)}}$

(E04) $\int x^{m}{\sqrt[{n}]{ax+b}}\;dx={\frac {n}{a^{m+1}}}\left(\sum _{k=0}^{m}\left(-1\right)^{k}{\frac {{m \choose k}b^{k}}{n\left(m-k+1\right)+1}}{\sqrt[{n}]{\left(ax+b\right)^{n\left(m-k+1\right)+1}}}\right)+C\qquad {\mbox{(dla }}n\geq 2,m\geq 0{\mbox{)}}$

(E05) $\int {\frac {dx}{\sqrt[{n}]{ax+b}}}={\frac {n}{a\left(n-1\right)}}{\sqrt[{n}]{\left(ax+b\right)^{n-1}}}+C\qquad {\mbox{(dla }}n\geq 2{\mbox{)}}$

(E06) $\int {\frac {x\;dx}{\sqrt[{n}]{ax+b}}}={\frac {n}{a^{2}}}\left({\frac {1}{2n-1}}{\sqrt[{n}]{\left(ax+b\right)^{2n-1}}}-{\frac {b}{n-1}}{\sqrt[{n}]{\left(ax+b\right)^{n-1}}}\right)+C\qquad {\mbox{(dla }}n\geq 2{\mbox{)}}$

(E07)$\int {\frac {x^{2}\;dx}{\sqrt[{n}]{ax+b}}}={\frac {n}{a^{3}}}\left({\frac {1}{3n-1}}{\sqrt[{n}]{\left(ax+b\right)^{3n-1}}}-{\frac {2b}{2n-1}}{\sqrt[{n}]{\left(ax+b\right)^{2n-1}}}+{\frac {b^{2}}{n-1}}{\sqrt[{n}]{\left(ax+b\right)^{n-1}}}\right)+C\qquad {\mbox{(dla }}n\geq 2{\mbox{)}}$

(E08) $\int {\frac {x^{m}\;dx}{\sqrt[{n}]{ax+b}}}={\frac {n}{a^{m+1}}}\left(\sum _{k=0}^{m}\left(-1\right)^{k}{\frac {{m \choose k}b^{k}}{n\left(m-k+1\right)-1}}{\sqrt[{n}]{\left(ax+b\right)^{n\left(m-k+1\right)-1}}}\right)+C\qquad {\mbox{(dla }}n\geq 2,m\geq 0{\mbox{)}}$

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