无理函数积分表

以下是部份無理函數的积分表。(书写时省略了不定积分结果中都含有的任意常数Cn) ∫ r d x = 1 2 [ x r + a 2 ln ⁡ ( x + r a ) ] {\displaystyle \int r\;dx={\frac {1}{2}}\left[xr+a^{2}\,\ln \left({\frac {x+r}{a}}\right)\right]} ∫ r 3 d x = 1 4 x r 3 + 1 8 3 a 2 x r + 3 8 a 4 ln ⁡ ( x + r a ) {\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)} ∫ r 5 d x = 1 6 x r 5 + 5 24 a 2 x r 3 + 5 16 a 4 x r + 5 16 a 6 ln ⁡ ( x + r a ) {\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^{6}\ln \left({\frac {x+r}{a}}\right)} ∫ x r d x = r 3 3 {\displaystyle \int xr\;dx={\frac {r^{3}}{3}}} ∫ x r 3 d x = r 5 5 {\displaystyle \int xr^{3}\;dx={\frac {r^{5}}{5}}} ∫ x r 2 n + 1 d x = r 2 n + 3 2 n + 3 {\displaystyle \int xr^{2n+1}\;dx={\frac {r^{2n+3}}{2n+3}}} ∫ x 2 r d x = x r 3 4 − a 2 x r 8 − a 4 8 ln ⁡ ( x + r a ) {\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)} ∫ x 2 r 3 d x = x r 5 6 − a 2 x r 3 24 − a 4 x r 16 − a 6 16 ln ⁡ ( x + r a ) {\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)} ∫ x 3 r d x = r 5 5 − a 2 r 3 3 {\displaystyle \int x^{3}r\;dx={\frac {r^{5}}{5}}-{\frac {a^{2}r^{3}}{3}}} ∫ x 3 r 3 d x = r 7 7 − a 2 r 5 5 {\displaystyle \int x^{3}r^{3}\;dx={\frac {r^{7}}{7}}-{\frac {a^{2}r^{5}}{5}}} ∫ x 3 r 2 n + 1 d x = r 2 n + 5 2 n + 5 − a 2 r 2 n + 3 2 n + 3 {\displaystyle \int x^{3}r^{2n+1}\;dx={\frac {r^{2n+5}}{2n+5}}-{\frac {a^{2}r^{2n+3}}{2n+3}}} ∫ x 4 r d x = x 3 r 3 6 − a 2 x r 3 8 + a 4 x r 16 + a 6 16 ln ⁡ ( x + r a ) {\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)} ∫ x 4 r 3 d x = x 3 r 5 8 − a 2 x r 5 16 + a 4 x r 3 64 + 3 a 6 x r 128 + 3 a 8 128 ln ⁡ ( x + r a ) {\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)} ∫ x 5 r d x = r 7 7 − 2 a 2 r 5 5 + a 4 r 3 3 {\displaystyle \int x^{5}r\;dx={\frac {r^{7}}{7}}-{\frac {2a^{2}r^{5}}{5}}+{\frac {a^{4}r^{3}}{3}}} ∫ x 5 r 3 d x = r 9 9 − 2 a 2 r 7 7 + a 4 r 5 5 {\displaystyle \int x^{5}r^{3}\;dx={\frac {r^{9}}{9}}-{\frac {2a^{2}r^{7}}{7}}+{\frac {a^{4}r^{5}}{5}}} ∫ x 5 r 2 n + 1 d x = r 2 n + 7 2 n + 7 − 2 a 2 r 2 n + 5 2 n + 5 + a 4 r 2 n + 3 2 n + 3 {\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}}} ∫ r d x x = r − a ln ⁡ | a + r x | = r − a sinh − 1 ⁡ a x {\displaystyle \int {\frac {r\;dx}{x}}=r-a\ln \left|{\frac {a+r}{x}}\right|=r-a\sinh ^{-1}{\frac {a}{x}}} ∫ r 3 d x x = r 3 3 + a 2 r − a 3 ln ⁡ | a + r x | {\displaystyle \int {\frac {r^{3}\;dx}{x}}={\frac {r^{3}}{3}}+a^{2}r-a^{3}\ln \left|{\frac {a+r}{x}}\right|} ∫ r 5 d x x = r 5 5 + a 2 r 3 3 + a 4 r − a 5 ln ⁡ | a + r x | {\displaystyle \int {\frac {r^{5}\;dx}{x}}={\frac {r^{5}}{5}}+{\frac {a^{2}r^{3}}{3}}+a^{4}r-a^{5}\ln \left|{\frac {a+r}{x}}\right|} ∫ r 7 d x x = r 7 7 + a 2 r 5 5 + a 4 r 3 3 + a 6 r − a 7 ln ⁡ | a + r x | {\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|} ∫ d x r = sinh − 1 ⁡ x a = ln ⁡ | x + r | {\displaystyle \int {\frac {dx}{r}}=\sinh ^{-1}{\frac {x}{a}}=\ln \left|x+r\right|} ∫ x d x r = r {\displaystyle \int {\frac {x\,dx}{r}}=r} ∫ x 2 d x r = x 2 r − a 2 2 sinh − 1 ⁡ x a = x 2 r − a 2 2 ln ⁡ | x + r | {\displaystyle \int {\frac {x^{2}\;dx}{r}}={\frac {x}{2}}r-{\frac {a^{2}}{2}}\,\sinh ^{-1}{\frac {x}{a}}={\frac {x}{2}}r-{\frac {a^{2}}{2}}\ln \left|x+r\right|} ∫ d x x r = − 1 a sinh − 1 ⁡ a x = − 1 a ln ⁡ | a + r x | {\displaystyle \int {\frac {dx}{xr}}=-{\frac {1}{a}}\,\sinh ^{-1}{\frac {a}{x}}=-{\frac {1}{a}}\ln \left|{\frac {a+r}{x}}\right|} 假設 ( x 2 > a 2 ) {\displaystyle (x^{2}>a^{2})} , ( x 2 < a 2 ) {\displaystyle (x^{2}<a^{2})} 請看下一節: ∫ x s d x = 1 3 s 3 {\displaystyle \int xs\;dx={\frac {1}{3}}s^{3}} ∫ s d x x = s − a cos − 1 ⁡ | a x | {\displaystyle \int {\frac {s\;dx}{x}}=s-a\cos ^{-1}\left|{\frac {a}{x}}\right|} ∫ d x s = ∫ d x x 2 − a 2 = ln ⁡ | x + s a | {\displaystyle \int {\frac {dx}{s}}=\int {\frac {dx}{\sqrt {x^{2}-a^{2}}}}=\ln \left|{\frac {x+s}{a}}\right|} 注意在 ln ⁡ | x + s a | = s g n ( x ) cosh − 1 ⁡ | x a | = 1 2 ln ⁡ ( x + s x − s ) {\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)} , 里 cosh − 1 ⁡ | x a | {\displaystyle \cosh ^{-1}\left|{\frac {x}{a}}\right|} 取正值. ∫ x d x s = s {\displaystyle \int {\frac {x\;dx}{s}}=s} ∫ x d x s 3 = − 1 s {\displaystyle \int {\frac {x\;dx}{s^{3}}}=-{\frac {1}{s}}} ∫ x d x s 5 = − 1 3 s 3 {\displaystyle \int {\frac {x\;dx}{s^{5}}}=-{\frac {1}{3s^{3}}}} ∫ x d x s 7 = − 1 5 s 5 {\displaystyle \int {\frac {x\;dx}{s^{7}}}=-{\frac {1}{5s^{5}}}} ∫ x d x s 2 n + 1 = − 1 ( 2 n − 1 ) s 2 n − 1 {\displaystyle \int {\frac {x\;dx}{s^{2n+1}}}=-{\frac {1}{(2n-1)s^{2n-1}}}} ∫ x 2 m d x s 2 n + 1 = − 1 2 n − 1 x 2 m − 1 s 2 n − 1 + 2 m − 1 2 n − 1 ∫ x 2 m − 2 d x s 2 n − 1 {\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}}}} ∫ x 2 d x s = x s 2 + a 2 2 ln ⁡ | x + s a | {\displaystyle \int {\frac {x^{2}\;dx}{s}}={\frac {xs}{2}}+{\frac {a^{2}}{2}}\ln \left|{\frac {x+s}{a}}\right|} ∫ x 2 d x s 3 = − x s + ln ⁡ | x + s a | {\displaystyle \int {\frac {x^{2}\;dx}{s^{3}}}=-{\frac {x}{s}}+\ln \left|{\frac {x+s}{a}}\right|} ∫ x 4 d x s = x 3 s 4 + 3 8 a 2 x s + 3 8 a 4 ln ⁡ | x + s a | {\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|} ∫ x 4 d x s 3 = x s 2 − a 2 x s + 3 2 a 2 ln ⁡ | x + s a | {\displaystyle \int {\frac {x^{4}\;dx}{s^{3}}}={\frac {xs}{2}}-{\frac {a^{2}x}{s}}+{\frac {3}{2}}a^{2}\ln \left|{\frac {x+s}{a}}\right|} ∫ x 4 d x s 5 = − x s − 1 3 x 3 s 3 + ln ⁡ | x + s a | {\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|} ∫ x 2 m d x s 2 n + 1 = ( − 1 ) n − m 1 a 2 ( n − m ) ∑ i = 0 n − m − 1 1 2 ( m + i ) + 1 ( n − m − 1 i ) x 2 ( m + i ) + 1 s 2 ( m + i ) + 1 ( n > m ≥ 0 ) {\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{)}}} ∫ d x s 3 = − 1 a 2 x s {\displaystyle \int {\frac {dx}{s^{3}}}=-{\frac {1}{a^{2}}}{\frac {x}{s}}} ∫ d x s 5 = 1 a 4 [ x s − 1 3 x 3 s 3 ] {\displaystyle \int {\frac {dx}{s^{5}}}={\frac {1}{a^{4}}}\left[{\frac {x}{s}}-{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}\right]} ∫ d x s 7 = − 1 a 6 [ x s − 2 3 x 3 s 3 + 1 5 x 5 s 5 ] {\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]} ∫ d x s 9 = 1 a 8 [ x s − 3 3 x 3 s 3 + 3 5 x 5 s 5 − 1 7 x 7 s 7 ] {\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]} ∫ x 2 d x s 5 = − 1 a 2 x 3 3 s 3 {\displaystyle \int {\frac {x^{2}\;dx}{s^{5}}}=-{\frac {1}{a^{2}}}{\frac {x^{3}}{3s^{3}}}} ∫ x 2 d x s 7 = 1 a 4 [ 1 3 x 3 s 3 − 1 5 x 5 s 5 ] {\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]} ∫ x 2 d x s 9 = − 1 a 6 [ 1 3 x 3 s 3 − 2 5 x 5 s 5 + 1 7 x 7 s 7 ] {\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]} ∫ t d x = 1 2 ( x t + a 2 sin − 1 ⁡ x a ) ( | x | ≤ | a | ) {\displaystyle \int t\;dx={\frac {1}{2}}\left(xt+a^{2}\sin ^{-1}{\frac {x}{a}}\right)\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}} ∫ x t d x = − 1 3 t 3 ( | x | ≤ | a | ) {\displaystyle \int xt\;dx=-{\frac {1}{3}}t^{3}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}} ∫ t d x x = t − a ln ⁡ | a + t x | ( | x | ≤ | a | ) {\displaystyle \int {\frac {t\;dx}{x}}=t-a\ln \left|{\frac {a+t}{x}}\right|\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}} ∫ d x t = sin − 1 ⁡ x a ( | x | ≤ | a | ) {\displaystyle \int {\frac {dx}{t}}=\sin ^{-1}{\frac {x}{a}}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}} ∫ x 2 d x t = − x 2 t + a 2 2 sin − 1 ⁡ x a ( | x | ≤ | a | ) {\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{)}}} ∫ t d x = 1 2 ( x t − sgn ⁡ x cosh − 1 ⁡ | x a | ) ( | x | ≥ | a | ) {\displaystyle \int t\;dx={\frac {1}{2}}\left(xt-\operatorname {sgn} x\,\cosh ^{-1}\left|{\frac {x}{a}}\right|\right)\qquad {\mbox{(}}|x|\geq |a|{\mbox{)}}} ∫ x n 1 − x 2 d x = ( x n + 1 − x n − 1 ) 1 − x 2 n + 1 + n − 1 n + 1 ∫ x n − 2 1 − x 2 d x {\displaystyle \int {x^{n}{\sqrt {1-x^{2}}}{\text{d}}x}={\frac {\left(x^{n+1}-x^{n-1}\right){\sqrt {1-x^{2}}}}{n+1}}+{\frac {n-1}{n+1}}\int {x^{n-2}{\sqrt {1-x^{2}}}{\text{d}}x}} ∫ d x a x 2 + b x + c = 1 a ln ⁡ | 2 a R + 2 a x + b | ( a > 0 ) {\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}={\frac {1}{\sqrt {a}}}\ln \left|2{\sqrt {aR}}+2ax+b\right|\qquad \qquad {\mbox{(}}a>0{\mbox{)}}} ∫ d x a x 2 + b x + c = 1 a sinh − 1 ⁡ 2 a x + b 4 a c − b 2 ( a > 0 ,  4 a c − b 2 > 0 ) {\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}={\frac {1}{\sqrt {a}}}\,\sinh ^{-1}{\frac {2ax+b}{\sqrt {4ac-b^{2}}}}\qquad \qquad {\mbox{(}}a>0{\mbox{, }}4ac-b^{2}>0{\mbox{)}}} ∫ d x a x 2 + b x + c = 1 a ln ⁡ | 2 a x + b | ( a > 0 ,  4 a c − b 2 = 0 ) {\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}={\frac {1}{\sqrt {a}}}\ln |2ax+b|\quad {\mbox{(}}a>0{\mbox{, }}4ac-b^{2}=0{\mbox{)}}} ∫ d x a x 2 + b x + c = − 1 − a arcsin ⁡ 2 a x + b b 2 − 4 a c ( a < 0 ,  4 a c − b 2 < 0 ) {\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}=-{\frac {1}{\sqrt {-a}}}\arcsin {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}\qquad \qquad {\mbox{(}}a<0{\mbox{, }}4ac-b^{2}<0{\mbox{)}}} ∫ d x ( a x 2 + b x + c ) 3 = 4 a x + 2 b ( 4 a c − b 2 ) R {\displaystyle \int {\frac {dx}{\sqrt {(ax^{2}+bx+c)^{3}}}}={\frac {4ax+2b}{(4ac-b^{2}){\sqrt {R}}}}} ∫ d x ( a x 2 + b x + c ) 5 = 4 a x + 2 b 3 ( 4 a c − b 2 ) R ( 1 R + 8 a 4 a c − b 2 ) {\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)} ∫ d x ( a x 2 + b x + c ) 2 n + 1 = 4 a x + 2 b ( 2 n − 1 ) ( 4 a c − b 2 ) R ( 2 n − 1 ) / 2 + 8 a ( n − 1 ) ( 2 n − 1 ) ( 4 a c − b 2 ) ∫ d x R ( 2 n − 1 ) / 2 {\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}}}} ∫ x d x a x 2 + b x + c = R a − b 2 a ∫ d x R {\displaystyle \int {\frac {x\;dx}{\sqrt {ax^{2}+bx+c}}}={\frac {\sqrt {R}}{a}}-{\frac {b}{2a}}\int {\frac {dx}{\sqrt {R}}}} ∫ x d x ( a x 2 + b x + c ) 3 = − 2 b x + 4 c ( 4 a c − b 2 ) R {\displaystyle \int {\frac {x\;dx}{\sqrt {(ax^{2}+bx+c)^{3}}}}=-{\frac {2bx+4c}{(4ac-b^{2}){\sqrt {R}}}}} ∫ x d x ( a x 2 + b x + c ) 2 n + 1 = − 1 ( 2 n − 1 ) a R ( 2 n − 1 ) / 2 − b 2 a ∫ d x R ( 2 n + 1 ) / 2 {\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}}}} ∫ d x x a x 2 + b x + c = − 1 c ln ⁡ ( 2 c R + b x + 2 c x ) {\displaystyle \int {\frac {dx}{x{\sqrt {ax^{2}+bx+c}}}}=-{\frac {1}{\sqrt {c}}}\ln \left({\frac {2{\sqrt {cR}}+bx+2c}{x}}\right)} ∫ d x x a x 2 + b x + c = − 1 c sinh − 1 ⁡ ( b x + 2 c | x | 4 a c − b 2 ) {\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)} ∫ d x x a x + b = − 2 b tanh − 1 ⁡ a x + b b {\displaystyle \int {\frac {dx}{x{\sqrt {ax+b}}}}\,=\,{\frac {-2}{\sqrt {b}}}\tanh ^{-1}{\sqrt {\frac {ax+b}{b}}}} ∫ a x + b x d x = 2 ( a x + b − b tanh − 1 ⁡ a x + b b ) {\displaystyle \int {\frac {\sqrt {ax+b}}{x}}\,dx\;=\;2\left({\sqrt {ax+b}}-{\sqrt {b}}\tanh ^{-1}{\sqrt {\frac {ax+b}{b}}}\right)} ∫ x n a x + b d x = 2 a ( 2 n + 1 ) ( x n a x + b − b n ∫ x n − 1 a x + b ) {\displaystyle \int {\frac {x^{n}}{\sqrt {ax+b}}}\,dx\;=\;{\frac {2}{a(2n+1)}}\left(x^{n}{\sqrt {ax+b}}-bn\int {\frac {x^{n-1}}{\sqrt {ax+b}}}\right)} ∫ x n a x + b d x = 2 2 n + 1 ( x n + 1 a x + b + b x n a x + b − n b ∫ x n − 1 a x + b d x ) {\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)}