قائمة بتكاملات التوابع غير المنطقة

توابع تحتوي ${\displaystyle{\displaystyle r={\sqrt{a^{2}+x^{2}}}}}$

${\displaystyle{\displaystyle\int r\;dx={\frac{1}{2}}\left(xr+a^{2}\,\ln\left({\frac{x+r}{a}}\right)\right)}}$

${\displaystyle{\displaystyle\int r^{3}\;dx={\frac{1}{4}}xr^{3}+{\frac{1}{8}}3a^{2}xr+{\frac{3}{8}}a^{4}\ln\left({\frac{x+r}{a}}\right)}}$

${\displaystyle{\displaystyle\int r^{5}\;dx={\frac{1}{6}}xr^{5}+{\frac{5}{24}}a^{2}xr^{3}+{\frac{5}{16}}a^{4}xr+{\frac{5}{16}}a^{5}\ln\left({\frac{x+r}{a}}\right)}}$

${\displaystyle{\displaystyle\int xr\;dx={\frac{r^{3}}{3}}}}$

${\displaystyle{\displaystyle\int xr^{3}\;dx={\frac{r^{5}}{5}}}}$

${\displaystyle{\displaystyle\int xr^{2n+1}\;dx={\frac{r^{2n+3}}{2n+3}}}}$

${\displaystyle{\displaystyle\int x^{2}r\;dx={\frac{xr^{3}}{4}}-{\frac{a^{2}xr}{8}}-{\frac{a^{4}}{8}}\ln\left({\frac{x+r}{a}}\right)}}$

${\displaystyle{\displaystyle\int x^{2}r^{3}\;dx={\frac{xr^{5}}{6}}-{\frac{a^{2}xr^{3}}{24}}-{\frac{a^{4}xr}{16}}-{\frac{a^{6}}{16}}\ln\left({\frac{x+r}{a}}\right)}}$

${\displaystyle{\displaystyle\int x^{3}r\;dx={\frac{r^{5}}{5}}-{\frac{a^{2}r^{3}}{3}}}}$

${\displaystyle{\displaystyle\int x^{3}r^{3}\;dx={\frac{r^{3}}{7}}-{\frac{a^{2}r^{5}}{5}}}}$

${\displaystyle{\displaystyle\int x^{3}r^{2n+1}\;dx={\frac{r^{2n+5}}{2n+5}}-{\frac{a^{3}r^{2n+3}}{2n+3}}}}$

${\displaystyle{\displaystyle\int x^{4}r\;dx={\frac{x^{3}r^{3}}{6}}-{\frac{a^{2}xr^{3}}{8}}-{\frac{a^{4}xr}{16}}+{\frac{a^{6}}{16}}\ln\left({\frac{x+r}{a}}\right)}}$

${\displaystyle{\displaystyle\int x^{4}r^{3}\;dx={\frac{x^{3}r^{5}}{8}}-{\frac{a^{2}xr^{5}}{16}}-{\frac{a^{4}xr^{3}}{64}}+{\frac{3a^{6}xr}{128}}+{\frac{3a^{8}}{128}}\ln\left({\frac{x+r}{a}}\right)}}$

${\displaystyle{\displaystyle\int x^{5}r\;dx={\frac{r^{7}}{7}}-{\frac{2a^{2}r^{5}}{5}}+{\frac{a^{4}r^{3}}{3}}}}$

${\displaystyle{\displaystyle\int x^{5}r^{3}\;dx={\frac{r^{9}}{9}}-{\frac{2a^{2}r^{7}}{7}}+{\frac{a^{4}r^{5}}{5}}}}$

${\displaystyle{\displaystyle\int x^{5}r^{2n+1}\;dx={\frac{r^{2n+7}}{2n+7}}-{\frac{2a^{2}r^{2n+5}}{2n+5}}+{\frac{a^{4}r^{2n+3}}{2n+3}}}}$

${\displaystyle{\displaystyle\int{\frac{r\;dx}{x}}=r-a\ln\left|{\frac{a+r}{x}}\right|=r-a\sinh^{-1}{\frac{a}{x}}}}$

${\displaystyle{\displaystyle\int{\frac{r^{3}\;dx}{x}}={\frac{r^{3}}{3}}+a^{2}r-a^{3}\ln\left|{\frac{a+r}{x}}\right|}}$

${\displaystyle{\displaystyle\int{\frac{r^{5}\;dx}{x}}={\frac{r^{5}}{5}}+{\frac{a^{3}r^{3}}{3}}+a^{4}r-a^{5}\ln\left|{\frac{a+r}{x}}\right|}}$

${\displaystyle{\displaystyle\int{\frac{r^{7}\;dx}{x}}={\frac{r^{7}}{7}}+{\frac{a^{2}r^{5}}{5}}+{\frac{a^{4}r^{3}}{3}}+a^{6}r-a^{7}\ln\left|{\frac{a+r}{x}}\right|}}$

${\displaystyle{\displaystyle\int{\frac{dx}{r}}=\sinh^{-1}{\frac{x}{a}}=\ln\left|x+r\right|}}$

${\displaystyle{\displaystyle\int{\frac{x\,dx}{r}}=r}}$

${\displaystyle{\displaystyle\int{\frac{x^{2}\;dx}{r}}={\frac{x}{2}}r-{\frac{a^{2}}{2}}\,\sinh^{-1}{\frac{a}{x}}={\frac{x}{2}}r-{\frac{a^{2}}{2}}\ln\left|x+r\right|}}$

${\displaystyle{\displaystyle\int{\frac{dx}{xr}}=-{\frac{1}{a}}\,\sinh^{-1}{\frac{a}{x}}=-{\frac{1}{a}}\ln\left|{\frac{a+r}{x}}\right|}}$

توابع تحتوي ${\displaystyle{\displaystyle s={\sqrt{x^{2}-a^{2}}}}}$

بفرض أن ${\displaystyle{\displaystyle(x^{2}>a^{2})}}$, for ${\displaystyle{\displaystyle(x^{2}<a^{2})}}$, انظر المقطع اللاحق :

${\displaystyle{\displaystyle\int xs\;dx={\frac{1}{3}}s^{3}}}$

${\displaystyle{\displaystyle\int{\frac{s\;dx}{x}}=s-a\cos^{-1}\left|{\frac{a}{x}}\right|}}$

${\displaystyle{\displaystyle\int{\frac{dx}{s}}=\int{\frac{dx}{\sqrt{x^{2}-a^{2}}}}=\ln\left|{\frac{x+s}{a}}\right|}}$

لاحظ أن ${\displaystyle{\displaystyle\ln\left|{\frac{x+s}{a}}\right|=\mathrm{sgn}(x)\cosh^{-1}\left|{\frac{x}{a}}\right|={\frac{1}{2}}\ln\left({\frac{x+s}{x-s}}\right)}}$, حيث القيمة الموجبة ل${\displaystyle{\displaystyle\cosh^{-1}\left|{\frac{x}{a}}\right|}}$ يجب أن تكون.

${\displaystyle{\displaystyle\int{\frac{x\;dx}{s}}=s}}$

${\displaystyle{\displaystyle\int{\frac{x\;dx}{s^{3}}}=-{\frac{1}{s}}}}$

${\displaystyle{\displaystyle\int{\frac{x\;dx}{s^{5}}}=-{\frac{1}{3s^{3}}}}}$

${\displaystyle{\displaystyle\int{\frac{x\;dx}{s^{7}}}=-{\frac{1}{5s^{5}}}}}$

${\displaystyle{\displaystyle\int{\frac{x\;dx}{s^{2n+1}}}=-{\frac{1}{(2n-1)s^{2n-1}}}}}$

${\displaystyle{\displaystyle\int{\frac{x^{2m}\;dx}{s^{2n+1}}}=-{\frac{1}{2n-1}}{\frac{x^{2m-1}}{s^{2n-1}}}+{\frac{2m-1}{2n-1}}\int{\frac{x^{2m-2}\;dx}{s^{2n-1}}}}}$

${\displaystyle{\displaystyle\int{\frac{x^{2}\;dx}{s}}={\frac{xs}{2}}+{\frac{a^{2}}{2}}\ln\left|{\frac{x+s}{a}}\right|}}$

${\displaystyle{\displaystyle\int{\frac{x^{2}\;dx}{s^{3}}}=-{\frac{x}{s}}+\ln\left|{\frac{x+s}{a}}\right|}}$

${\displaystyle{\displaystyle\int{\frac{x^{4}\;dx}{s}}={\frac{x^{3}s}{4}}+{\frac{3}{8}}a^{2}xs+{\frac{3}{8}}a^{4}\ln\left|{\frac{x+s}{a}}\right|}}$

${\displaystyle{\displaystyle\int{\frac{x^{4}\;dx}{s^{3}}}={\frac{xs}{2}}-{\frac{a^{2}x}{s}}+{\frac{3}{2}}\ln\left|{\frac{x+s}{a}}\right|}}$

${\displaystyle{\displaystyle\int{\frac{x^{4}\;dx}{s^{5}}}=-{\frac{x}{s}}-{\frac{1}{3}}{\frac{x^{3}}{s^{3}}}+\ln\left|{\frac{x+s}{a}}\right|}}$

${\displaystyle{\displaystyle\int{\frac{x^{2m}\;dx}{s^{2n+1}}}=(-1)^{n-m}{\frac{1}{a^{2(n-m)}}}\sum_{i=0}^{n-m-1}{\frac{1}{2(m+i)+1}}{n-m-1\choose i}{\frac{x^{2(m+i)+1}}{s^{2(m+i)+1}}}\qquad{\mbox{(}}n>m\geq 0{\mbox{)}}}}$

${\displaystyle{\displaystyle\int{\frac{dx}{s^{3}}}=-{\frac{1}{a^{2}}}{\frac{x}{s}}}}$

${\displaystyle{\displaystyle\int{\frac{dx}{s^{5}}}={\frac{1}{a^{4}}}\left[{\frac{x}{s}}-{\frac{1}{3}}{\frac{x^{3}}{s^{3}}}\right]}}$

${\displaystyle{\displaystyle\int{\frac{dx}{s^{7}}}=-{\frac{1}{a^{6}}}\left[{\frac{x}{s}}-{\frac{2}{3}}{\frac{x^{3}}{s^{3}}}+{\frac{1}{5}}{\frac{x^{5}}{s^{5}}}\right]}}$

${\displaystyle{\displaystyle\int{\frac{dx}{s^{9}}}={\frac{1}{a^{8}}}\left[{\frac{x}{s}}-{\frac{3}{3}}{\frac{x^{3}}{s^{3}}}+{\frac{3}{5}}{\frac{x^{5}}{s^{5}}}-{\frac{1}{7}}{\frac{x^{7}}{s^{7}}}\right]}}$

${\displaystyle{\displaystyle\int{\frac{x^{2}\;dx}{s^{5}}}=-{\frac{1}{a^{2}}}{\frac{x^{3}}{3s^{3}}}}}$

${\displaystyle{\displaystyle\int{\frac{x^{2}\;dx}{s^{7}}}={\frac{1}{a^{4}}}\left[{\frac{1}{3}}{\frac{x^{3}}{s^{3}}}-{\frac{1}{5}}{\frac{x^{5}}{s^{5}}}\right]}}$

${\displaystyle{\displaystyle\int{\frac{x^{2}\;dx}{s^{9}}}=-{\frac{1}{a^{6}}}\left[{\frac{1}{3}}{\frac{x^{3}}{s^{3}}}-{\frac{2}{5}}{\frac{x^{5}}{s^{5}}}+{\frac{1}{7}}{\frac{x^{7}}{s^{7}}}\right]}}$

توابع تحتوي ${\displaystyle{\displaystyle t={\sqrt{a^{2}-x^{2}}}}}$

${\displaystyle{\displaystyle\int t\;dx={\frac{1}{2}}\left(xt+a^{2}\sin^{-1}{\frac{x}{a}}\right)\qquad{\mbox{(}}|x|\leq|a|{\mbox{)}}}}$

${\displaystyle{\displaystyle\int xt\;dx=-{\frac{1}{3}}t^{3}\qquad{\mbox{(}}|x|\leq|a|{\mbox{)}}}}$

${\displaystyle{\displaystyle\int{\frac{t\;dx}{x}}=t-a\ln\left|{\frac{a+t}{x}}\right|\qquad{\mbox{(}}|x|\leq|a|{\mbox{)}}}}$

${\displaystyle{\displaystyle\int{\frac{dx}{t}}=\sin^{-1}{\frac{x}{a}}\qquad{\mbox{(}}|x|\leq|a|{\mbox{)}}}}$

${\displaystyle{\displaystyle\int{\frac{x^{2}\;dx}{t}}=-{\frac{x}{2}}t+{\frac{a^{2}}{2}}\sin^{-1}{\frac{x}{a}}\qquad{\mbox{(}}|x|\leq|a|{\mbox{)}}}}$

${\displaystyle{\displaystyle\int t\;dx={\frac{1}{2}}\left(xt-\operatorname{sgn}x\,\cosh^{-1}\left|{\frac{x}{a}}\right|\right)\qquad{\mbox{(for }}|x|\geq|a|{\mbox{)}}}}$

توابع تحتوي ${\displaystyle{\displaystyle R^{1/2}={\sqrt{ax^{2}+bx+c}}}}$

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

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

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

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

${\displaystyle{\displaystyle\int{\frac{dx}{\sqrt{(ax^{2}+bx+c)^{3}}}}={\frac{4ax+2b}{(4ac-b^{2}){\sqrt{R}}}}}}$

${\displaystyle{\displaystyle\int{\frac{dx}{\sqrt{(ax^{2}+bx+c)^{5}}}}={\frac{4ax+2b}{3(4ac-b^{2}){\sqrt{R}}}}\left({\frac{1}{R}}+{\frac{8a}{4ac-b^{2}}}\right)}}$

${\displaystyle{\displaystyle\int{\frac{dx}{\sqrt{(ax^{2}+bx+c)^{2n+1}}}}={\frac{4ax+2b}{(2n-1)(4ac-b^{2})R^{(2n-1)/2}}}+{\frac{8a(n-1)}{(2n-1)(4ac-b^{2})}}\int{\frac{dx}{R^{(2n-1)/2}}}}}$

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

${\displaystyle{\displaystyle\int{\frac{x\;dx}{\sqrt{(ax^{2}+bx+c)^{3}}}}=-{\frac{2bx+4c}{(4ac-b^{2}){\sqrt{R}}}}}}$

${\displaystyle{\displaystyle\int{\frac{x\;dx}{\sqrt{(ax^{2}+bx+c)^{2n+1}}}}=-{\frac{1}{(2n-1)aR^{(2n-1)/2}}}-{\frac{b}{2a}}\int{\frac{dx}{R^{(2n+1)/2}}}}}$

${\displaystyle{\displaystyle\int{\frac{dx}{x{\sqrt{ax^{2}+bx+c}}}}=-{\frac{1}{\sqrt{c}}}\ln\left({\frac{2{\sqrt{cR}}+bx+2c}{x}}\right)}}$

${\displaystyle{\displaystyle\int{\frac{dx}{x{\sqrt{ax^{2}+bx+c}}}}=-{\frac{1}{\sqrt{c}}}\sinh^{-1}\left({\frac{bx+2c}{|x|{\sqrt{4ac-b^{2}}}}}\right)}}$

توابع تحتوي ${\displaystyle{\displaystyle R^{1/2}={\sqrt{ax+b}}}}$

${\displaystyle{\displaystyle\int{\frac{dx}{x{\sqrt{ax+b}}}}\,=\,{\frac{-2}{\sqrt{b}}}\tanh^{-1}{\sqrt{\frac{ax+b}{b}}}}}$

${\displaystyle{\displaystyle\int{\frac{\sqrt{ax+b}}{x}}\,dx\;=\;2\left({\sqrt{ax+b}}-{\sqrt{b}}\tanh^{-1}{\sqrt{\frac{ax+b}{b}}}\right)}}$

${\displaystyle{\displaystyle\int{\frac{x^{n}}{\sqrt{ax+b}}}\,dx\;=\;{\frac{2}{a}}\left(x^{n+1}{\sqrt{ax+b}}+bx^{n}{\sqrt{ax+b}}-nb\int{\frac{x^{n-1}}{\sqrt{ax+b}}}\right)}}$

${\displaystyle{\displaystyle\int x^{n}{\sqrt{ax+b}}\,dx\;=\;{\frac{2}{2n+1}}\left(x^{n+1}{\sqrt{ax+b}}+bx^{n}{\sqrt{ax+b}}-nb\int x^{n-1}{\sqrt{ax+b}}\,dx\right)}}$