قائمة بتكاملات الدوال النسبية

هذه قائمة بتكاملات الدوال النسبية:

${\displaystyle{\displaystyle\int(ax+b)^{n}dx={\frac{(ax+b)^{n+1}}{a(n+1)}}\qquad{\mbox{(for }}n\neq-1{\mbox{)}}\,\!}}$

${\displaystyle{\displaystyle\int{\frac{dx}{ax+b}}={\frac{1}{a}}\ln\left|ax+b\right|}}$

${\displaystyle{\displaystyle\int x(ax+b)^{n}dx={\frac{a(n+1)x-b}{a^{2}(n+1)(n+2)}}(ax+b)^{n+1}\qquad{\mbox{(for }}n\not\in\{-1,-2\}{\mbox{)}}}}$

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

${\displaystyle{\displaystyle\int{\frac{x}{(ax+b)^{2}}}dx={\frac{b}{a^{2}(ax+b)}}+{\frac{1}{a^{2}}}\ln\left|ax+b\right|}}$

${\displaystyle{\displaystyle\int{\frac{x}{(ax+b)^{n}}}dx={\frac{a(1-n)x-b}{a^{2}(n-1)(n-2)(ax+b)^{n-1}}}\qquad{\mbox{(for }}n\not\in\{-1,-2\}{\mbox{)}}}}$

${\displaystyle{\displaystyle\int{\frac{x^{2}}{ax+b}}dx={\frac{1}{a^{3}}}\left({\frac{(ax+b)^{2}}{2}}-2b(ax+b)+b^{2}\ln\left|ax+b\right|\right)}}$

${\displaystyle{\displaystyle\int{\frac{x^{2}}{(ax+b)^{2}}}dx={\frac{1}{a^{3}}}\left(ax+b-2b\ln\left|ax+b\right|-{\frac{b^{2}}{ax+b}}\right)}}$

${\displaystyle{\displaystyle\int{\frac{x^{2}}{(ax+b)^{3}}}dx={\frac{1}{a^{3}}}\left(\ln\left|ax+b\right|+{\frac{2b}{ax+b}}-{\frac{b^{2}}{2(ax+b)^{2}}}\right)}}$

${\displaystyle{\displaystyle\int{\frac{x^{2}}{(ax+b)^{n}}}dx={\frac{1}{a^{3}}}\left(-{\frac{1}{(n-3)(ax+b)^{n-3}}}+{\frac{2b}{(n-2)(a+b)^{n-2}}}-{\frac{b^{2}}{(n-1)(ax+b)^{n-1}}}\right)\qquad{\mbox{(for }}n\not\in\{1,2,3\}{\mbox{)}}}}$

${\displaystyle{\displaystyle\int{\frac{dx}{x(ax+b)}}=-{\frac{1}{b}}\ln\left|{\frac{ax+b}{x}}\right|}}$

${\displaystyle{\displaystyle\int{\frac{dx}{x^{2}(ax+b)}}=-{\frac{1}{bx}}+{\frac{a}{b^{2}}}\ln\left|{\frac{ax+b}{x}}\right|}}$

${\displaystyle{\displaystyle\int{\frac{dx}{x^{2}(ax+b)^{2}}}=-a\left({\frac{1}{b^{2}(ax+b)}}+{\frac{1}{ab^{2}x}}-{\frac{2}{b^{3}}}\ln\left|{\frac{ax+b}{x}}\right|\right)}}$

${\displaystyle{\displaystyle\int{\frac{dx}{x^{2}+a^{2}}}={\frac{1}{a}}\arctan{\frac{x}{a}}\,\!}}$

${\displaystyle{\displaystyle\int{\frac{dx}{x^{2}-a^{2}}}=-{\frac{1}{a}}\,\mathrm{artanh}{\frac{x}{a}}={\frac{1}{2a}}\ln{\frac{a-x}{a+x}}\qquad{\mbox{(for }}|x|<|a|{\mbox{)}}\,\!}}$

${\displaystyle{\displaystyle\int{\frac{dx}{x^{2}-a^{2}}}=-{\frac{1}{a}}\,\mathrm{arcoth}{\frac{x}{a}}={\frac{1}{2a}}\ln{\frac{x-a}{x+a}}\qquad{\mbox{(for }}|x|>|a|{\mbox{)}}\,\!}}$

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

${\displaystyle{\displaystyle\int{\frac{dx}{ax^{2}+bx+c}}={\frac{2}{\sqrt{b^{2}-4ac}}}\,\mathrm{artanh}{\frac{2ax+b}{\sqrt{b^{2}-4ac}}}={\frac{1}{\sqrt{b^{2}-4ac}}}\ln\left|{\frac{2ax+b-{\sqrt{b^{2}-4ac}}}{2ax+b+{\sqrt{b^{2}-4ac}}}}\right|\qquad{\mbox{(for }}4ac-b^{2}<0{\mbox{)}}}}$

${\displaystyle{\displaystyle\int{\frac{x}{ax^{2}+bx+c}}dx={\frac{1}{2a}}\ln\left|ax^{2}+bx+c\right|-{\frac{b}{2a}}\int{\frac{dx}{ax^{2}+bx+c}}}}$

${\displaystyle{\displaystyle\int{\frac{mx+n}{ax^{2}+bx+c}}dx={\frac{m}{2a}}\ln\left|ax^{2}+bx+c\right|+{\frac{2an-bm}{a{\sqrt{4ac-b^{2}}}}}\arctan{\frac{2ax+b}{\sqrt{4ac-b^{2}}}}\qquad{\mbox{(for }}4ac-b^{2}>0{\mbox{)}}}}$

${\displaystyle{\displaystyle\int{\frac{mx+n}{ax^{2}+bx+c}}dx={\frac{m}{2a}}\ln\left|ax^{2}+bx+c\right|+{\frac{2an-bm}{a{\sqrt{b^{2}-4ac}}}}\,\mathrm{artanh}{\frac{2ax+b}{\sqrt{b^{2}-4ac}}}\qquad{\mbox{(for }}4ac-b^{2}<0{\mbox{)}}}}$

${\displaystyle{\displaystyle\int{\frac{dx}{(ax^{2}+bx+c)^{n}}}={\frac{2ax+b}{(n-1)(4ac-b^{2})(ax^{2}+bx+c)^{n-1}}}+{\frac{(2n-3)2a}{(n-1)(4ac-b^{2})}}\int{\frac{dx}{(ax^{2}+bx+c)^{n-1}}}\,\!}}$

${\displaystyle{\displaystyle\int{\frac{x}{(ax^{2}+bx+c)^{n}}}dx={\frac{bx+2c}{(n-1)(4ac-b^{2})(ax^{2}+bx+c)^{n-1}}}-{\frac{b(2n-3)}{(n-1)(4ac-b^{2})}}\int{\frac{dx}{(ax^{2}+bx+c)^{n-1}}}\,\!}}$

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