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Aşağıdaki liste integrallerini vermektedir ax b ndx ax b n 1a n 1 n 1 for displaystyle int ax b n dx frac ax b n 1 a

Rasyonel fonksiyonların integralleri

Rasyonel fonksiyonların integralleri
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Aşağıdaki liste integrallerini vermektedir

∫(ax+b)ndx=(ax+b)n+1a(n+1)( n≠−1(for ){\displaystyle \int (ax+b)^{n}dx={\frac {(ax+b)^{n+1}}{a(n+1)}}\qquad {\text{( }}n\neq -1{\text{(for )}}\,\!}{\displaystyle \int (ax+b)^{n}dx={\frac {(ax+b)^{n+1}}{a(n+1)}}\qquad {\text{( }}n\neq -1{\text{(for )}}\,\!}
∫dxax+b=1aln⁡|ax+b|{\displaystyle \int {\frac {dx}{ax+b}}={\frac {1}{a}}\ln \left|ax+b\right|}{\displaystyle \int {\frac {dx}{ax+b}}={\frac {1}{a}}\ln \left|ax+b\right|}
∫x(ax+b)ndx=a(n+1)x−ba2(n+1)(n+2)(ax+b)n+1(for n∉{−2,−1}){\displaystyle \int x(ax+b)^{n}dx={\frac {a(n+1)x-b}{a^{2}(n+1)(n+2)}}(ax+b)^{n+1}\qquad {\text{(for }}n\not \in \{-2,-1\}{\text{)}}}{\displaystyle \int x(ax+b)^{n}dx={\frac {a(n+1)x-b}{a^{2}(n+1)(n+2)}}(ax+b)^{n+1}\qquad {\text{(for }}n\not \in \{-2,-1\}{\text{)}}}
∫xax+bdx=xa−ba2ln⁡|ax+b|{\displaystyle \int {\frac {x}{ax+b}}dx={\frac {x}{a}}-{\frac {b}{a^{2}}}\ln \left|ax+b\right|}{\displaystyle \int {\frac {x}{ax+b}}dx={\frac {x}{a}}-{\frac {b}{a^{2}}}\ln \left|ax+b\right|}
∫x(ax+b)2dx=ba2(ax+b)+1a2ln⁡|ax+b|{\displaystyle \int {\frac {x}{(ax+b)^{2}}}dx={\frac {b}{a^{2}(ax+b)}}+{\frac {1}{a^{2}}}\ln \left|ax+b\right|}{\displaystyle \int {\frac {x}{(ax+b)^{2}}}dx={\frac {b}{a^{2}(ax+b)}}+{\frac {1}{a^{2}}}\ln \left|ax+b\right|}
∫x(ax+b)ndx=a(1−n)x−ba2(n−1)(n−2)(ax+b)n−1(for n∉{1,2}){\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 {\text{(for }}n\not \in \{1,2\}{\text{)}}}{\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 {\text{(for }}n\not \in \{1,2\}{\text{)}}}
∫x2ax+bdx=1a3((ax+b)22−2b(ax+b)+b2ln⁡|ax+b|){\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 \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)}
∫x2(ax+b)2dx=1a3(ax+b−2bln⁡|ax+b|−b2ax+b){\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 \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)}
∫x2(ax+b)3dx=1a3(ln⁡|ax+b|+2bax+b−b22(ax+b)2){\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 \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)}
∫x2(ax+b)ndx=1a3(−1(n−3)(ax+b)n−3+2b(n−2)(ax+b)n−2−b2(n−1)(ax+b)n−1)(for n∉{1,2,3}){\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)(ax+b)^{n-2}}}-{\frac {b^{2}}{(n-1)(ax+b)^{n-1}}}\right)\qquad {\text{(for }}n\not \in \{1,2,3\}{\text{)}}}{\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)(ax+b)^{n-2}}}-{\frac {b^{2}}{(n-1)(ax+b)^{n-1}}}\right)\qquad {\text{(for }}n\not \in \{1,2,3\}{\text{)}}}
∫dxx(ax+b)=−1bln⁡|ax+bx|{\displaystyle \int {\frac {dx}{x(ax+b)}}=-{\frac {1}{b}}\ln \left|{\frac {ax+b}{x}}\right|}{\displaystyle \int {\frac {dx}{x(ax+b)}}=-{\frac {1}{b}}\ln \left|{\frac {ax+b}{x}}\right|}
∫dxx2(ax+b)=−1bx+ab2ln⁡|ax+bx|{\displaystyle \int {\frac {dx}{x^{2}(ax+b)}}=-{\frac {1}{bx}}+{\frac {a}{b^{2}}}\ln \left|{\frac {ax+b}{x}}\right|}{\displaystyle \int {\frac {dx}{x^{2}(ax+b)}}=-{\frac {1}{bx}}+{\frac {a}{b^{2}}}\ln \left|{\frac {ax+b}{x}}\right|}
∫dxx2(ax+b)2=−a(1b2(ax+b)+1ab2x−2b3ln⁡|ax+bx|){\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 \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)}
∫dxx2+a2=1aarctan⁡xa{\displaystyle \int {\frac {dx}{x^{2}+a^{2}}}={\frac {1}{a}}\arctan {\frac {x}{a}}\,\!}{\displaystyle \int {\frac {dx}{x^{2}+a^{2}}}={\frac {1}{a}}\arctan {\frac {x}{a}}\,\!}
∫dxx2−a2=−1aartanh⁡xa=12aln⁡a−xa+x(for |x|<|a|){\displaystyle \int {\frac {dx}{x^{2}-a^{2}}}=-{\frac {1}{a}}\operatorname {artanh} {\frac {x}{a}}={\frac {1}{2a}}\ln {\frac {a-x}{a+x}}\qquad {\text{(for }}|x|<|a|{\text{)}}\,\!}{\displaystyle \int {\frac {dx}{x^{2}-a^{2}}}=-{\frac {1}{a}}\operatorname {artanh} {\frac {x}{a}}={\frac {1}{2a}}\ln {\frac {a-x}{a+x}}\qquad {\text{(for }}|x|<|a|{\text{)}}\,\!}
∫dxx2−a2=−1aarcoth⁡xa=12aln⁡x−ax+a(for |x|>|a|){\displaystyle \int {\frac {dx}{x^{2}-a^{2}}}=-{\frac {1}{a}}\operatorname {arcoth} {\frac {x}{a}}={\frac {1}{2a}}\ln {\frac {x-a}{x+a}}\qquad {\text{(for }}|x|>|a|{\text{)}}\,\!}{\displaystyle \int {\frac {dx}{x^{2}-a^{2}}}=-{\frac {1}{a}}\operatorname {arcoth} {\frac {x}{a}}={\frac {1}{2a}}\ln {\frac {x-a}{x+a}}\qquad {\text{(for }}|x|>|a|{\text{)}}\,\!}
∫dxax2+bx+c=24ac−b2arctan⁡2ax+b4ac−b2(for 4ac−b2>0){\displaystyle \int {\frac {dx}{ax^{2}+bx+c}}={\frac {2}{\sqrt {4ac-b^{2}}}}\arctan {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}\qquad {\text{(for }}4ac-b^{2}>0{\text{)}}}{\displaystyle \int {\frac {dx}{ax^{2}+bx+c}}={\frac {2}{\sqrt {4ac-b^{2}}}}\arctan {\frac {2ax+b}{\sqrt {4ac-b^{2}}}}\qquad {\text{(for }}4ac-b^{2}>0{\text{)}}}
∫dxax2+bx+c=2b2−4acartanh⁡2ax+bb2−4ac=1b2−4acln⁡|2ax+b−b2−4ac2ax+b+b2−4ac|(for 4ac−b2<0){\displaystyle \int {\frac {dx}{ax^{2}+bx+c}}={\frac {2}{\sqrt {b^{2}-4ac}}}\operatorname {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 {\text{(for }}4ac-b^{2}<0{\text{)}}}{\displaystyle \int {\frac {dx}{ax^{2}+bx+c}}={\frac {2}{\sqrt {b^{2}-4ac}}}\operatorname {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 {\text{(for }}4ac-b^{2}<0{\text{)}}}
∫dxax2+bx+c=−22ax+b(for 4ac−b2=0){\displaystyle \int {\frac {dx}{ax^{2}+bx+c}}=-{\frac {2}{2ax+b}}\qquad {\text{(for }}4ac-b^{2}=0{\text{)}}}{\displaystyle \int {\frac {dx}{ax^{2}+bx+c}}=-{\frac {2}{2ax+b}}\qquad {\text{(for }}4ac-b^{2}=0{\text{)}}}
∫xax2+bx+cdx=12aln⁡|ax2+bx+c|−b2a∫dxax2+bx+c{\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 \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}}}
∫mx+nax2+bx+cdx=m2aln⁡|ax2+bx+c|+2an−bma4ac−b2arctan⁡2ax+b4ac−b2(for 4ac−b2>0){\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 {\text{(for }}4ac-b^{2}>0{\text{)}}}{\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 {\text{(for }}4ac-b^{2}>0{\text{)}}}
∫mx+nax2+bx+cdx=m2aln⁡|ax2+bx+c|+2an−bmab2−4acartanh⁡2ax+bb2−4ac(for 4ac−b2<0){\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}}}}\operatorname {artanh} {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}\qquad {\text{(for }}4ac-b^{2}<0{\text{)}}}{\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}}}}\operatorname {artanh} {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}\qquad {\text{(for }}4ac-b^{2}<0{\text{)}}}
∫mx+nax2+bx+cdx=m2aln⁡|ax2+bx+c|−2an−bma(2ax+b)(For 4ac−b2=0){\displaystyle \int {\frac {mx+n}{ax^{2}+bx+c}}dx={\frac {m}{2a}}\ln \left|ax^{2}+bx+c\right|-{\frac {2an-bm}{a(2ax+b)}}\qquad {\text{(For }}4ac-b^{2}=0{\text{)}}}{\displaystyle \int {\frac {mx+n}{ax^{2}+bx+c}}dx={\frac {m}{2a}}\ln \left|ax^{2}+bx+c\right|-{\frac {2an-bm}{a(2ax+b)}}\qquad {\text{(For }}4ac-b^{2}=0{\text{)}}}
∫dx(ax2+bx+c)n=2ax+b(n−1)(4ac−b2)(ax2+bx+c)n−1+(2n−3)2a(n−1)(4ac−b2)∫dx(ax2+bx+c)n−1{\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 \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}}}\,\!}
∫x(ax2+bx+c)ndx=bx+2c(n−1)(4ac−b2)(ax2+bx+c)n−1−b(2n−3)(n−1)(4ac−b2)∫dx(ax2+bx+c)n−1{\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 \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}}}\,\!}
∫dxx(ax2+bx+c)=12cln⁡|x2ax2+bx+c|−b2c∫dxax2+bx+c{\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}}}{\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}}}
∫x2r+y2r=r{\displaystyle \int {\frac {x^{2}}{r}}+{\frac {y^{2}}{r}}=r}{\displaystyle \int {\frac {x^{2}}{r}}+{\frac {y^{2}}{r}}=r}
∫ |x|+|y|=|n|{\displaystyle \int \ |x|+|y|=|n|}{\displaystyle \int \ |x|+|y|=|n|}

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Asagidaki liste integrallerini vermektedir ax b ndx ax b n 1a n 1 n 1 for displaystyle int ax b n dx frac ax b n 1 a n 1 qquad text n neq 1 text for dxax b 1aln ax b displaystyle int frac dx ax b frac 1 a ln left ax b right x ax b ndx a n 1 x ba2 n 1 n 2 ax b n 1 for n 2 1 displaystyle int x ax b n dx frac a n 1 x b a 2 n 1 n 2 ax b n 1 qquad text for n not in 2 1 text xax bdx xa ba2ln ax b displaystyle int frac x ax b dx frac x a frac b a 2 ln left ax b right x ax b 2dx ba2 ax b 1a2ln ax b displaystyle int frac x ax b 2 dx frac b a 2 ax b frac 1 a 2 ln left ax b right x ax b ndx a 1 n x ba2 n 1 n 2 ax b n 1 for n 1 2 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 text for n not in 1 2 text x2ax bdx 1a3 ax b 22 2b ax b b2ln ax b 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 x2 ax b 2dx 1a3 ax b 2bln ax b b2ax b 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 x2 ax b 3dx 1a3 ln ax b 2bax b b22 ax b 2 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 x2 ax b ndx 1a3 1 n 3 ax b n 3 2b n 2 ax b n 2 b2 n 1 ax b n 1 for n 1 2 3 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 ax b n 2 frac b 2 n 1 ax b n 1 right qquad text for n not in 1 2 3 text dxx ax b 1bln ax bx displaystyle int frac dx x ax b frac 1 b ln left frac ax b x right dxx2 ax b 1bx ab2ln ax bx displaystyle int frac dx x 2 ax b frac 1 bx frac a b 2 ln left frac ax b x right dxx2 ax b 2 a 1b2 ax b 1ab2x 2b3ln ax bx 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 dxx2 a2 1aarctan xa displaystyle int frac dx x 2 a 2 frac 1 a arctan frac x a dxx2 a2 1aartanh xa 12aln a xa x for x lt a displaystyle int frac dx x 2 a 2 frac 1 a operatorname artanh frac x a frac 1 2a ln frac a x a x qquad text for x lt a text dxx2 a2 1aarcoth xa 12aln x ax a for x gt a displaystyle int frac dx x 2 a 2 frac 1 a operatorname arcoth frac x a frac 1 2a ln frac x a x a qquad text for x gt a text dxax2 bx c 24ac b2arctan 2ax b4ac b2 for 4ac b2 gt 0 displaystyle int frac dx ax 2 bx c frac 2 sqrt 4ac b 2 arctan frac 2ax b sqrt 4ac b 2 qquad text for 4ac b 2 gt 0 text dxax2 bx c 2b2 4acartanh 2ax bb2 4ac 1b2 4acln 2ax b b2 4ac2ax b b2 4ac for 4ac b2 lt 0 displaystyle int frac dx ax 2 bx c frac 2 sqrt b 2 4ac operatorname 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 text for 4ac b 2 lt 0 text dxax2 bx c 22ax b for 4ac b2 0 displaystyle int frac dx ax 2 bx c frac 2 2ax b qquad text for 4ac b 2 0 text xax2 bx cdx 12aln ax2 bx c b2a dxax2 bx c 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 mx nax2 bx cdx m2aln ax2 bx c 2an bma4ac b2arctan 2ax b4ac b2 for 4ac b2 gt 0 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 text for 4ac b 2 gt 0 text mx nax2 bx cdx m2aln ax2 bx c 2an bmab2 4acartanh 2ax bb2 4ac for 4ac b2 lt 0 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 operatorname artanh frac 2ax b sqrt b 2 4ac qquad text for 4ac b 2 lt 0 text mx nax2 bx cdx m2aln ax2 bx c 2an bma 2ax b For 4ac b2 0 displaystyle int frac mx n ax 2 bx c dx frac m 2a ln left ax 2 bx c right frac 2an bm a 2ax b qquad text For 4ac b 2 0 text dx ax2 bx c n 2ax b n 1 4ac b2 ax2 bx c n 1 2n 3 2a n 1 4ac b2 dx ax2 bx c n 1 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 x ax2 bx c ndx bx 2c n 1 4ac b2 ax2 bx c n 1 b 2n 3 n 1 4ac b2 dx ax2 bx c n 1 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 dxx ax2 bx c 12cln x2ax2 bx c b2c dxax2 bx c 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 x2r y2r r displaystyle int frac x 2 r frac y 2 r r x y n displaystyle int x y n Ayrica bakinizIntegral tablosu

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