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规则[编辑 | 编辑源代码]
∫ c ⋅ f ( x ) d x = c ⋅ ∫ f ( x ) d x {\displaystyle \int c\cdot f(x)\mathrm {d} x=c\cdot \int f(x)\mathrm {d} x}
∫ ( f ( x ) ± g ( x ) ) d x = ∫ f ( x ) d x ± ∫ g ( x ) d x {\displaystyle \int {\big (}f(x)\pm g(x){\big )}\mathrm {d} x=\int f(x)\mathrm {d} x\pm \int g(x)\mathrm {d} x}
∫ f ( g ( x ) ) g ′ ( x ) d x = ∫ f ( u ) d u = F ( u ) + C = F ( g ( x ) ) + C {\displaystyle \int f(g(x))g'(x)\mathrm {d} x=\int f(u)\mathrm {d} u=F(u)+C=F(g(x))+C} 其中 F ′ = f {\displaystyle F'=f}
∫ u d v = u v − ∫ v d u {\displaystyle \int u\,dv=uv-\int v\,du}
幂[编辑 | 编辑源代码]
∫ d x = x + C {\displaystyle \int \mathrm {d} x=x+C}
∫ a d x = a x + C {\displaystyle \int a\,\mathrm {d} x=ax+C}
∫ x n d x = x n + 1 n + 1 + C ( for n ≠ − 1 ) {\displaystyle \int x^{n}\mathrm {d} x={\frac {x^{n+1}}{n+1}}+C\qquad ({\text{for }}n\neq -1)}
∫ d x x = ln | x | + C {\displaystyle \int {\frac {\mathrm {d} x}{x}}=\ln |x|+C}
∫ d x a x + b = ln | a x + b | a + C ( for a ≠ 0 ) {\displaystyle \int {\frac {\mathrm {d} x}{ax+b}}={\frac {\ln |ax+b|}{a}}+C\qquad ({\text{for }}a\neq 0)}
三角函数[编辑 | 编辑源代码]
基本三角函数[编辑 | 编辑源代码]
∫ sin ( x ) d x = − cos ( x ) + C {\displaystyle \int \sin(x)\mathrm {d} x=-\cos(x)+C}
∫ cos ( x ) d x = sin ( x ) + C {\displaystyle \int \cos(x)\mathrm {d} x=\sin(x)+C}
∫ tan ( x ) d x = ln | sec ( x ) | + C {\displaystyle \int \tan(x)\mathrm {d} x=\ln |\sec(x)|+C}
∫ sin 2 ( x ) d x = ∫ 1 − cos ( 2 x ) 2 d x = x 2 − sin ( 2 x ) 4 + C {\displaystyle \int \sin ^{2}(x)\mathrm {d} x=\int {\frac {1-\cos(2x)}{2}}\mathrm {d} x={\frac {x}{2}}-{\frac {\sin(2x)}{4}}+C}
∫ cos 2 ( x ) d x = ∫ 1 + cos ( 2 x ) 2 d x = x 2 + sin ( 2 x ) 4 + C {\displaystyle \int \cos ^{2}(x)\mathrm {d} x=\int {\frac {1+\cos(2x)}{2}}\mathrm {d} x={\frac {x}{2}}+{\frac {\sin(2x)}{4}}+C}
∫ tan 2 ( x ) d x = tan ( x ) − x + C {\displaystyle \int \tan ^{2}(x)\mathrm {d} x=\tan(x)-x+C}
倒数三角函数[编辑 | 编辑源代码]
∫ sec ( x ) d x = ln | sec ( x ) + tan ( x ) | + C = ln | tan ( x 2 + π 4 ) | + C = 2 a r t a n h ( tan ( x 2 ) ) + C {\displaystyle \int \sec(x)\mathrm {d} x=\ln {\Big |}\sec(x)+\tan(x){\Big |}+C=\ln \left|\tan \left({\frac {x}{2}}+{\frac {\pi }{4}}\right)\right|+C=2\mathrm {artanh} \left(\tan \left({\frac {x}{2}}\right)\right)+C}
∫ csc ( x ) d x = − ln | csc ( x ) + cot ( x ) | + C = ln | tan ( x 2 ) | + C {\displaystyle \int \csc(x)\mathrm {d} x=-\ln {\Big |}\csc(x)+\cot(x){\Big |}+C=\ln \left|\tan \left({\frac {x}{2}}\right)\right|+C}
∫ cot ( x ) d x = ln | sin ( x ) | + C {\displaystyle \int \cot(x)\mathrm {d} x=\ln |\sin(x)|+C}
∫ sec 2 ( a x ) d x = tan ( a x ) a + C {\displaystyle \int \sec ^{2}(ax)\mathrm {d} x={\frac {\tan(ax)}{a}}+C}
∫ csc 2 ( a x ) d x = − cot ( a x ) a + C {\displaystyle \int \csc ^{2}(ax)\mathrm {d} x=-{\frac {\cot(ax)}{a}}+C}
∫ cot 2 ( a x ) d x = − x − cot ( a x ) a + C {\displaystyle \int \cot ^{2}(ax)\mathrm {d} x=-x-{\frac {\cot(ax)}{a}}+C}
∫ sec ( x ) tan ( x ) d x = sec ( x ) + C {\displaystyle \int \sec(x)\tan(x)\mathrm {d} x=\sec(x)+C}
∫ sec ( x ) csc ( x ) d x = ln | tan ( x ) | + C {\displaystyle \int \sec(x)\csc(x)\mathrm {d} x=\ln |\tan(x)|+C}
降阶公式[编辑 | 编辑源代码]
∫ sin n ( x ) d x = − sin n − 1 ( x ) cos ( x ) n + n − 1 n ∫ sin n − 2 ( x ) d x + C ( for n > 0 ) {\displaystyle \int \sin ^{n}(x)\mathrm {d} x=-{\frac {\sin ^{n-1}(x)\cos(x)}{n}}+{\frac {n-1}{n}}\int \sin ^{n-2}(x)\mathrm {d} x+C\qquad ({\text{for }}n>0)}
∫ cos n ( x ) d x = − cos n − 1 ( x ) sin ( x ) n + n − 1 n ∫ cos n − 2 ( x ) d x + C ( for n > 0 ) {\displaystyle \int \cos ^{n}(x)\mathrm {d} x=-{\frac {\cos ^{n-1}(x)\sin(x)}{n}}+{\frac {n-1}{n}}\int \cos ^{n-2}(x)\mathrm {d} x+C\qquad ({\text{for }}n>0)}
∫ tan n ( x ) d x = tan n − 1 ( x ) ( n − 1 ) − ∫ tan n − 2 ( x ) d x + C ( for n ≠ 1 ) {\displaystyle \int \tan ^{n}(x)\mathrm {d} x={\frac {\tan ^{n-1}(x)}{(n-1)}}-\int \tan ^{n-2}(x)\mathrm {d} x+C\qquad ({\text{for }}n\neq 1)}
∫ sec n ( x ) d x = sec n − 1 ( x ) sin ( x ) n − 1 + n − 2 n − 1 ∫ sec n − 2 ( x ) d x + C ( for n ≠ 1 ) {\displaystyle \int \sec ^{n}(x)\mathrm {d} x={\frac {\sec ^{n-1}(x)\sin(x)}{n-1}}+{\frac {n-2}{n-1}}\int \sec ^{n-2}(x)\mathrm {d} x+C\qquad ({\text{for }}n\neq 1)}
∫ csc n ( x ) d x = − csc n − 1 ( x ) cos ( x ) n − 1 + n − 2 n − 1 ∫ csc n − 2 ( x ) d x + C ( for n ≠ 1 ) {\displaystyle \int \csc ^{n}(x)\mathrm {d} x=-{\frac {\csc ^{n-1}(x)\cos(x)}{n-1}}+{\frac {n-2}{n-1}}\int \csc ^{n-2}(x)\mathrm {d} x+C\qquad ({\text{for }}n\neq 1)}
∫ cot n ( x ) d x = −
a 2 ∫ x n sin ( a x ) d x = n x n − 1 sin ( a x ) − a x n cos ( a x ) − n ( n − 1 ) ∫ x n − 2 sin ( a x ) d x {\displaystyle a^{2}\int x^{n}\sin(ax)\mathrm {d} x=nx^{n-1}\sin(ax)-ax^{n}\cos(ax)-n(n-1)\int x^{n-2}\sin(ax)\mathrm {d} x}
a 2 ∫ x n cos ( a x ) d x = a x n sin ( a x ) + n x n − 1 cos ( a x ) − n ( n − 1 ) ∫ x n − 2 cos ( a x ) d x {\displaystyle a^{2}\int x^{n}\cos(ax)\mathrm {d} x=ax^{n}\sin(ax)+nx^{n-1}\cos(ax)-n(n-1)\int x^{n-2}\cos(ax)\mathrm {d} x}
明确形式[编辑 | 编辑源代码]
∫ sin n ( x ) d x = − cos ( x ) 2 F 1 ( 1 2 , 1 − n 2 ; 3 2 ; cos 2 ( x ) ) + C {\displaystyle \int \sin ^{n}(x)\mathrm {d} x=-\cos(x)_{2}F_{1}\left({\frac {1}{2}},{\frac {1-n}{2}};{\frac {3}{2}};\cos ^{2}(x)\right)+C}
∫ cos n ( x ) d x = − 1 n + 1 s g n ( sin ( x ) ) cos n + 1 ( x ) 2 F 1 ( 1 2 , n + 1 2 ; n + 3 2 ; cos 2 ( x ) ) + C ( for n ≠ − 1 ) {\displaystyle \int \cos ^{n}(x)\mathrm {d} x=-{\frac {1}{n+1}}\mathrm {sgn} (\sin(x))\cos ^{n+1}(x)_{2}F_{1}\left({\frac {1}{2}},{\frac {n+1}{2}};{\frac {n+3}{2}};\cos ^{2}(x)\right)+C\qquad ({\text{for }}n\neq -1)}
∫ tan n ( x ) d x = 1 n + 1 tan n + 1 ( x ) 2 F 1 ( 1 , n + 1 2 ; n + 3 2 ; − tan 2 ( x ) ) + C ( for n ≠ − 1 ) {\displaystyle \int \tan ^{n}(x)\mathrm {d} x={\frac {1}{n+1}}\tan ^{n+1}(x)_{2}F_{1}\left(1,{\frac {n+1}{2}};{\frac {n+3}{2}};-\tan ^{2}(x)\right)+C\qquad ({\text{for }}n\neq -1)}
∫ csc n ( x ) d x = − cos ( x ) 2 F 1 ( 1 2 , n + 1 2 ; 3 2 ; cos 2 ( x ) ) + C {\displaystyle \int \csc ^{n}(x)\mathrm {d} x=-\cos(x)_{2}F_{1}\left({\frac {1}{2}},{\frac {n+1}{2}};{\frac {3}{2}};\cos ^{2}(x)\right)+C}
∫ sec n ( x ) d x = sin ( x ) 2 F 1 ( 1 2 , n + 1 2 ; 3 2 ; sin 2 ( x ) ) + C {\displaystyle \int \sec ^{n}(x)\mathrm {d} x=\sin(x)_{2}F_{1}\left({\frac {1}{2}},{\frac {n+1}{2}};{\frac {3}{2}};\sin ^{2}(x)\right)+C}
∫ cot n ( x ) d x = − 1 n + 1 cot n + 1 ( x ) 2 F 1 ( 1 , n + 1 2 ; n + 3 2 ; − cot 2 ( x ) ) + C ( for n ≠ − 1 ) {\displaystyle \int \cot ^{n}(x)\mathrm {d} x=-{\frac {1}{n+1}}\cot ^{n+1}(x)_{2}F_{1}\left(1,{\frac {n+1}{2}};{\frac {n+3}{2}};-\cot ^{2}(x)\right)+C\qquad ({\text{for }}n\neq -1)}
其中 2 F 1 {\displaystyle {}_{2}F_{1}} 是超几何函数， s g n {\displaystyle \mathrm {sgn} } 是符号函数。
反三角函数[edit | edit source]
∫ d x 1 − x 2 = arcsin ( x ) + C {\displaystyle \int {\frac {\mathrm {d} x}{\sqrt {1-x^{2}}}}=\arcsin(x)+C}
∫ d x a 2 − x 2 = arcsin ( x a ) + C ( for a ≠ 0 ) {\displaystyle \int {\frac {\mathrm {d} x}{\sqrt {a^{2}-x^{2}}}}=\arcsin \left({\tfrac {x}{a}}\right)+C\qquad ({\text{for }}a\neq 0)}
∫ d x 1 + x 2 = arctan ( x ) + C {\displaystyle \int {\frac {\mathrm {d} x}{1+x^{2}}}=\arctan(x)+C}
∫ d x a 2 + x 2 = arctan ( x a ) a + C ( for a ≠ 0 ) {\displaystyle \int {\frac {\mathrm {d} x}{a^{2}+x^{2}}}={\frac {\arctan \left({\tfrac {x}{a}}\right)}{a}}+C\qquad ({\text{for }}a\neq 0)}
指数函数和对数函数[edit | edit source]
∫ e x d x = e x + C {\displaystyle \int e^{x}\mathrm {d} x=e^{x}+C}
∫ e a x d x = e a x a + C ( for a ≠ 0 ) {\displaystyle \int e^{ax}\mathrm {d} x={\frac {e^{ax}}{a}}+C\qquad ({\text{for }}a\neq 0)}
∫ a x d x = a x ln ( a ) + C ( for a > 0 , a ≠ 1 ) {\displaystyle \int a^{x}\mathrm {d} x={\frac {a^{x}}{\ln(a)}}+C\qquad ({\text{for }}a>0,a\neq 1)}
∫ ln ( x ) d x = x ln ( x ) − x + C {\displaystyle \int \ln(x)\mathrm {d} x=x\ln(x)-x+C}
∫ e x sin ( x ) d x = e x 2 ( sin ( x ) − cos ( x ) ) + C {\displaystyle \int e^{x}\sin(x)\mathrm {d} x={\frac {e^{x}}{2}}(\sin(x)-\cos(x))+C}
∫ e x cos ( x ) d x = e x 2 ( sin ( x ) + cos ( x ) ) + C {\displaystyle \int e^{x}\cos(x)\mathrm {d} x={\frac {e^{x}}{2}}(\sin(x)+\cos(x))+C}
降阶公式[edit | edit source]
∫ x n e a x d x = 1 a x n e a x − n a ∫ x n − 1 e a x d x {\displaystyle \int x^{n}e^{ax}\mathrm {d} x={\frac {1}{a}}x^{n}e^{ax}-{\frac {n}{a}}\int x^{n-1}e^{ax}\mathrm {d} x}
反三角函数[edit | edit source]
∫ arcsin ( x ) d x = x arcsin ( x ) + 1 − x 2 + C {\displaystyle \int \arcsin(x)\mathrm {d} x=x\arcsin(x)+{\sqrt {1-x^{2}}}+C}
∫ arccos ( x ) d x = x arccos ( x ) − 1 − x 2 + C {\displaystyle \int \arccos(x)\mathrm {d} x=x\arccos(x)-{\sqrt {1-x^{2}}}+C}
∫ arctan ( x ) d x = x arctan ( x ) − 1 2 ln | 1 + x 2 | + C {\displaystyle \int \arctan(x)\mathrm {d} x=x\arctan(x)-{\frac {1}{2}}\ln |1+x^{2}|+C}
∫ arccsc ( x ) d x = x arccsc ( x ) + ln | x + x 1 − 1 x 2 | + C {\displaystyle \int \operatorname {arccsc}(x)\mathrm {d} x=x\operatorname {arccsc}(x)+\ln \left|x+x{\sqrt {1-{\frac {1}{x^{2}}}}}\right|+C}
∫ arcsec ( x ) d x = x arcsec ( x ) − ln | x + x 1 − 1 x 2 | + C {\displaystyle \int \operatorname {arcsec}(x)\mathrm {d} x=x\operatorname {arcsec}(x)-\ln \left|x+x{\sqrt {1-{\frac {1}{x^{2}}}}}\right|+C}
∫ arccot ( x ) d x = x arccot ( x ) + 1 2 ln | 1 + x 2 | + C {\displaystyle \int \operatorname {arccot}(x)\mathrm {d} x=x\operatorname {arccot}(x)+{\frac {1}{2}}\ln |1+x^{2}|+C}
双曲函数[edit | edit source]
∫ sinh ( x ) d x = − i ∫ sin ( i x ) d x = cos ( i x ) + C = cosh ( x ) + C {\displaystyle \int \sinh(x)\mathrm {d} x=-i\int \sin(ix)\mathrm {d} x=\cos(ix)+C=\cosh(x)+C}
∫ cosh ( x ) d x = ∫ cos ( i x ) d x = − i sin ( i x ) + C = sinh ( x ) + C {\displaystyle \int \cosh(x)\mathrm {d} x=\int \cos(ix)\mathrm {d} x=-i\sin(ix)+C=\sinh(x)+C}
∫ tanh ( x ) d x = − i ∫ tan ( i x ) d x = log | cos ( i x ) | + C = log | cosh ( x ) | + C {\displaystyle \int \tanh(x)\mathrm {d} x=-i\int \tan(ix)\mathrm {d} x=\log \left|\cos(ix)\right|+C=\log \left|\cosh(x)\right|+C}
倒数[编辑 | 编辑源代码]
∫ c s c h ( x ) d x = i ∫ csc ( i x ) d x = log | − i tan ( i x 2 ) | + C = log | tanh ( x 2 ) | + C {\displaystyle \int \mathrm {csch} (x)\mathrm {d} x=i\int \csc(ix)\mathrm {d} x=\log \left|-i\tan \left({\frac {ix}{2}}\right)\right|+C=\log \left|\tanh \left({\frac {x}{2}}\right)\right|+C}
∫ s e c h ( x ) d x = ∫ sec ( i x ) d x = 2 a r t a n h ( − i tan ( x 2 i ) ) + C = 2 arctan ( tanh ( x 2 ) ) + C {\displaystyle \int \mathrm {sech} (x)\mathrm {d} x=\int \sec(ix)\mathrm {d} x=2\mathrm {artanh} \left(-i\tan \left({\frac {x}{2}}i\right)\right)+C=2\arctan \left(\tanh \left({\frac {x}{2}}\right)\right)+C}
∫ c o t h ( x ) d x = i ∫ cot ( i x ) d x = log | − i sin ( i x ) | + C = log | sinh ( x ) | + C {\displaystyle \int \mathrm {coth} (x)\mathrm {d} x=i\int \cot(ix)\mathrm {d} x=\log \left|-i\sin(ix)\right|+C=\log \left|\sinh(x)\right|+C}
反函数[编辑 | 编辑源代码]
∫ a r s i n h ( x ) d x = x a r s i n h ( x ) − x 2 + 1 + C {\displaystyle \int \mathrm {arsinh} (x)\mathrm {d} x=x\mathrm {arsinh} (x)-{\sqrt {x^{2}+1}}+C}
∫ a r c o s h ( x ) d x = x a r c o s h ( x ) − x 2 − 1 + C {\displaystyle \int \mathrm {arcosh} (x)\mathrm {d} x=x\mathrm {arcosh} (x)-{\sqrt {x^{2}-1}}+C}
∫ a r t a n h ( x ) d x = x a r t a n h ( x ) + 1 2 ln ( 1 − x 2 ) + C {\displaystyle \int \mathrm {artanh} (x)\mathrm {d} x=x\mathrm {artanh} (x)+{\frac {1}{2}}\ln(1-x^{2})+C}
∫ a r c s c h ( x ) d x = x a r c s c h ( x ) + | a r s i n h ( x ) | + C {\displaystyle \int \mathrm {arcsch} (x)\mathrm {d} x=x\mathrm {arcsch} (x)+|\mathrm {arsinh} (x)|+C}
∫ a r s e c h ( x ) d x = x a r s e c h ( x ) + arcsin ( x ) + C {\displaystyle \int \mathrm {arsech} (x)\mathrm {d} x=x\mathrm {arsech} (x)+\arcsin(x)+C}
∫ a r t a n h ( x ) d x = x a r c o t h ( x ) + 1 2 ln ( x 2 − 1 ) + C {\displaystyle \int \mathrm {artanh} (x)\mathrm {d} x=x\mathrm {arcoth} (x)+{\frac {1}{2}}\ln(x^{2}-1)+C}
其他[edit | edit source]
∫ | f ( x ) | d x = s g n ( f ( x ) ) ∫ f ( x ) d x {\displaystyle \int |f(x)|\mathrm {d} x=\mathrm {sgn} (f(x))\int f(x)\mathrm {d} x} ，其中 s g n {\displaystyle \mathrm {sgn} } 是符号函数。
定积分[edit | edit source]
∫ [ 0 , 1 ] n ∏ i = 1 n d x i 1 − ∏ i = 1 n x i = ζ ( n ) for all integers n > 1 {\displaystyle \int _{[0,1]^{n}}{\frac {\prod _{i=1}^{n}\mathrm {d} x_{i}}{1-\prod _{i=1}^{n}x_{i}}}=\zeta (n){\text{ for all integers }}n>1} ，其中 ζ {\displaystyle \zeta } 是黎曼ζ函数。
∫ − ∞ ∞ e − x 2 d x = π {\displaystyle \int _{-\infty }^{\infty }e^{-x^{2}}\mathrm {d} x={\sqrt {\pi }}}
∫ 0 1 t u − 1 ( 1 − t ) v − 1 d t = β ( u , v ) = Γ ( u ) Γ ( v ) Γ ( u + v ) {\displaystyle \int _{0}^{1}t^{u-1}(1-t)^{v-1}\mathrm {d} t=\beta (u,v)={\frac {\Gamma (u)\Gamma (v)}{\Gamma (u+v)}}} ，其中 Γ {\displaystyle \Gamma } 是伽马函数。
∫ 0 ∞ t s − 1 e − t d t = Γ ( s ) {\displaystyle \int _{0}^{\infty }t^{s-1}e^{-t}\mathrm {d} t=\Gamma (s)}
∫ 0 2 π e u cos θ d θ = 2 π I 0 ( u ) {\displaystyle \int _{0}^{2\pi }e^{u\cos \theta }\mathrm {d} \theta =2\pi I_{0}(u)} ，其中 I 0 {\displaystyle I_{0}} 是第一类修正贝塞尔函数。
∫ 0 ∞ sin ( x ) x d x = π 2 {\displaystyle \int _{0}^{\infty }{\frac {\sin(x)}{x}}\mathrm {d} x={\frac {\pi }{2}}}
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