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  • 积分常用公式表
    千次阅读
    2021-04-20 14:24:31

    微分、导数和积分公式对照表

    序号微分公式导数公式积分公式
    |幂函数
    1 d ( x μ ) = μ x μ − 1   d x \mathbf{d}(x^\mu)=\mu x^{\mu-1}\ \mathbf{d}x d(xμ)=μxμ1 dx ( x μ ) ′ = μ x μ − 1 (x^\mu)'=\mu x^{\mu-1} (xμ)=μxμ1 ∫ x μ d x = x μ + 1 μ + 1 + C \int x^\mu \mathbf{d}x=\cfrac{x^{\mu+1}}{\mu+1}+C xμdx=μ+1xμ+1+C
    |指数函数
    2 d ( a x ) = a x ln ⁡ a   d x \mathbf{d}(a^x)=a^x \ln a\ \mathbf{d}x d(ax)=axlna dx ( a x ) ′ = a x ln ⁡ a (a^x)'=a^x \ln a (ax)=axlna ∫ a x d x = a x ln ⁡ a + C \int a^x\mathbf{d}x=\cfrac{a^x}{\ln a}+C axdx=lnaax+C
    3 d ( e x ) = e x   d x \mathbf{d}(e^x)=e^x\ \mathbf{d}x d(ex)=ex dx ( e x ) ′ = e x (e^x)'=e^x (ex)=ex ∫ e x d x = e x + C \int e^x \mathbf{d}x=e^x+C exdx=ex+C
    |对数函数
    4 d ( log ⁡ a x ) = 1 x ln ⁡ a d x \mathbf{d}(\log_ax)=\cfrac{1}{x\ln a}\mathbf{d}x d(logax)=xlna1dx ( log ⁡ a x ) ′ = 1 x ln ⁡ a (\log_ax)'=\cfrac{1}{x\ln a} (logax)=xlna1
    5 d ( ln ⁡ x ) = 1 x d x \mathbf{d}(\ln x)=\cfrac{1}{x}\mathbf{d}x d(lnx)=x1dx ( ln ⁡ x ) ′ = 1 x (\ln x)'=\cfrac{1}{x} (lnx)=x1 ∫ 1 x d x = ln ⁡ ∣ x ∣ + C \int \cfrac{1}{x} \mathbf{d}x=\ln \vert x \vert+C x1dx=lnx+C
    |三角函数
    6 d ( sin ⁡ x ) = cos ⁡ x   d x \mathbf{d}(\sin x)=\cos x\ \mathbf{d}x d(sinx)=cosx dx ( sin ⁡ x ) ′ = cos ⁡ x (\sin x)'=\cos x (sinx)=cosx ∫ cos ⁡ x d x = sin ⁡ x + C \int \cos x\mathbf{d}x=\sin x+C cosxdx=sinx+C
    7 d ( cos ⁡ x ) = − sin ⁡ x   d x \mathbf{d}(\cos x)=-\sin x\ \mathbf{d}x d(cosx)=sinx dx ( cos ⁡ x ) ′ = − sin ⁡ x (\cos x)'=-\sin x (cosx)=sinx ∫ sin ⁡ x d x = − cos ⁡ x + C \int \sin x \mathbf{d}x=-\cos x+C sinxdx=cosx+C
    8 d ( tan ⁡ x ) = sec ⁡ 2 x   d x \mathbf{d}(\tan x)=\sec^2x\ \mathbf{d}x d(tanx)=sec2x dx ( tan ⁡ x ) ′ = sec ⁡ 2 x (\tan x)'=\sec^2x (tanx)=sec2x ∫ sec ⁡ 2 x d x = ∫ 1 cos ⁡ 2 x = tan ⁡ x + C \int \sec^2x\mathbf{d}x=\int \cfrac{1}{\cos^2x}=\tan x+C sec2xdx=cos2x1=tanx+C
    |
    9 d ( cot ⁡ x ) = − csc ⁡ 2 x   d x \mathbf{d}(\cot x)=-\csc^2x\ \mathbf{d}x d(cotx)=csc2x dx ( cot ⁡ x ) ′ = − csc ⁡ 2 x (\cot x)'=-\csc^2x (cotx)=csc2x ∫ csc ⁡ 2 x d x = ∫ 1 sin ⁡ 2 x = − cot ⁡ x + C \int \csc^2x\mathbf{d}x=\int \cfrac{1}{\sin^2x}=-\cot x+C csc2xdx=sin2x1=cotx+C
    10 d ( sec ⁡ x ) = sec ⁡ x   tan ⁡ x   d x \mathbf{d}(\sec x)=\sec x\ \tan x\ \mathbf{d}x d(secx)=secx tanx dx ( sec ⁡ x ) ′ = sec ⁡ x tan ⁡ x (\sec x)'=\sec x\tan x (secx)=secxtanx ∫ sec ⁡ x tan ⁡ x d x = sec ⁡ x + C \int \sec x \tan x\mathbf{d}x=\sec x+C secxtanxdx=secx+C
    11 d ( csc ⁡ x ) = − csc ⁡ x cot ⁡ x   d x \mathbf{d}(\csc x)=-\csc x\cot x\ \mathbf{d}x d(cscx)=cscxcotx dx ( csc ⁡ x ) ′ = − csc ⁡ x cot ⁡ x (\csc x)'=-\csc x\cot x (cscx)=cscxcotx ∫ csc ⁡ x cot ⁡ x d x = − csc ⁡ x + C \int \csc x \cot x \mathbf{d}x=-\csc x+C cscxcotxdx=cscx+C
    |反三角函数
    12 d ( arcsin ⁡ x ) = 1 1 − x 2 d x \mathbf{d}(\arcsin x)=\cfrac{1}{\sqrt{1-x^2}}\mathbf{d}x d(arcsinx)=1x2 1dx ( arcsin ⁡ x ) ′ = 1 1 − x 2 (\arcsin x)'=\cfrac{1}{\sqrt{1-x^2}} (arcsinx)=1x2 1 ∫ 1 1 − x 2 = arcsin ⁡ x + C \int \cfrac{1}{\sqrt{1-x^2}}=\arcsin x+C 1x2 1=arcsinx+C
    13 d ( arccos ⁡ x ) = − 1 1 − x 2 d x \mathbf{d}(\arccos x)=-\cfrac{1}{\sqrt{1-x^2}}\mathbf{d}x d(arccosx)=1x2 1dx ( arccos ⁡ x ) ′ = − 1 1 − x 2 (\arccos x)'=-\cfrac{1}{\sqrt{1-x^2}} (arccosx)=1x2 1
    14 d ( arctan ⁡ x ) = 1 1 + x 2 d x \mathbf{d}(\arctan x)=\cfrac{1}{1+x^2}\mathbf{d}x d(arctanx)=1+x21dx ( arctan ⁡ x ) ′ = 1 1 + x 2 (\arctan x)'=\cfrac{1}{1+x^2} (arctanx)=1+x21 ∫ 1 1 + x 2 = arctan ⁡ x + C \int \cfrac{1}{1+x^2}=\arctan x+C 1+x21=arctanx+C
    15 d ( a r c c o t   x ) = − 1 1 + x 2 d x \mathbf{d}(arccot\ x)=-\cfrac{1}{1+x^2}\mathbf{d}x d(arccot x)=1+x21dx ( a r c c o t x ) ′ = − 1 1 + x 2 (arccot x)'=-\cfrac{1}{1+x^2} (arccotx)=1+x21

    积分续表

    ∫ k d x = k x + C , ( k 、 C 是 常 数 ) \int k \mathbf{d}x=kx+C,(k、C是常数) kdx=kx+C,kC

    ∫ tan ⁡ x d x = − ln ⁡ ∣ cos ⁡ x ∣ + C \int \tan x\mathbf{d}x=-\ln |\cos x|+C tanxdx=lncosx+C

    ∫ cot ⁡ x d x = ln ⁡ ∣ sin ⁡ x ∣ + C \int \cot x\mathbf{d}x=\ln |\sin x|+C cotxdx=lnsinx+C

    ∫ sec ⁡ x d x = ln ⁡ ∣ sec ⁡ x + tan ⁡ x ∣ + C \int \sec x\mathbf{d}x=\ln|\sec x+\tan x|+C secxdx=lnsecx+tanx+C

    ∫ csc ⁡ x d x = ln ⁡ ∣ csc ⁡ x − cot ⁡ x ∣ + C \int \csc x\mathbf{d}x=\ln|\csc x-\cot x|+C cscxdx=lncscxcotx+C

    ∫ 1 a 2 + x 2 d x = 1 a arctan ⁡ x a + C \int \cfrac{1}{a^2+x^2}\mathbf{d}x=\cfrac{1}{a}\arctan \cfrac{x}{a}+C a2+x21dx=a1arctanax+C

    ∫ 1 x 2 − a 2 d x = 1 2 a ln ⁡ ∣ x − a x + a ∣ + C \int \cfrac{1}{x^2-a^2}\mathbf{d}x=\cfrac{1}{2a}\ln \vert \dfrac{x-a}{x+a} \vert + C x2a21dx=2a1lnx+axa+C

    ∫ 1 a 2 − x 2 d x = arcsin ⁡ x a + C \int \cfrac{1}{\sqrt{a^2-x^2}}\mathbf{d}x=\arcsin \cfrac{x}{a}+C a2x2 1dx=arcsinax+C

    ∫ 1 x 2 + a 2 d x = ln ⁡ ∣ x + x 2 + a 2 ∣ + C \int \cfrac{1}{\sqrt{x^2+a^2}}\mathbf{d}x=\ln |x+\sqrt{x^2+a^2}|+C x2+a2 1dx=lnx+x2+a2 +C

    ∫ 1 x 2 − a 2 d x = ln ⁡ ∣ x + x 2 − a 2 ∣ + C \int \cfrac{1}{\sqrt{x^2-a^2}}\mathbf{d}x=\ln|x+\sqrt{x^2-a^2}|+C x2a2 1dx=lnx+x2a2 +C

    更多相关内容
  • 一、含有ax+b的积分 二、含有根号下ax+b的积分 三、含有x^2 ± a^2的积分 四、含有a*x^2+b的积分 五、含有a*x^2+bx+c的积分 六、含有根号下x^2+a^2的积分 七、含有根号下x^2-...

    一、含有ax+b的积分

    二、含有根号下ax+b的积分

     

     

    三、含有x^2 ± a^2的积分

     

    四、含有a*x^2+b的积分

     

     

     

    五、含有a*x^2+bx+c的积分

     

    六、含有根号下x^2+a^2的积分

     

     

    七、含有根号下x^2-a^2的积分

     

     

    八、含有根号下a^2-x^2的积分 

     

     

     

    九、含有根号下±ax^2+bx+c的积分 

     

     

     十、含有(x-a)(b-x)的积分

     

    十一、 含有三角函数的积分

     

     

     

    十二、含有反三角函数的积分

     

     

     

     十三、含有指数函数的积分

     

    展开全文
  • 数学分析 积分常用积分公式

    千次阅读 2020-07-29 22:21:38
    一.含有xnx^nxn的形式 二.含有ax+bax+bax+b的形式 三.含有ax2+bax^2+bax2+b的形式 四.含有a2±x2(a>0)a^2±x^2(a>0)a2±x2(a>0)的形式 五.含有ax2+bx+c(a>0)ax^2+bx+c(a>...六....七....八....0)\s

    一.含有 x n x^n xn的形式

    1. ∫ x n d x = x n + 1 n + 1 + C   ( n ≠ − 1 ) 1.\int x^ndx=\frac{x^{n+1}}{n+1}+C\,(n≠-1) 1.xndx=n+1xn+1+C(n=1)

    2. ∫ d x x = l n   ∣ x ∣ + C 2.\int\frac{dx}{x}=ln\,|x|+C 2.xdx=lnx+C

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    二.含有 a x + b ax+b ax+b的形式

    1. ∫ d x a x + b = 1 a ln ⁡ ∣ a x + b ∣ + C 1.\int{\frac{dx}{ax+b}}=\frac{1}{a}\ln{\mid ax+b \mid}+C 1.ax+bdx=a1lnax+b+C

    2. ∫ ( a x + b ) n d x = 1 a ( n + 1 ) ( a x + b ) n + 1 + C ( n ≠ − 1 ) 2.\int{(ax+b)^ndx}=\frac{1}{a(n+1)}(ax+b)^{n+1}+C(n\neq-1) 2.(ax+b)ndx=a(n+1)1(ax+b)n+1+C(n=1)

    3. ∫ x a x + b d x = 1 a 2 ( a x − b ln ⁡ ∣ a x + b ∣ ) + C 3.\int{\frac{x}{ax+b}dx}=\frac{1}{a^2}(ax-b\ln{\mid ax+b \mid})+C 3.ax+bxdx=a21(axblnax+b)+C

    4. ∫ x 2 a x + b d x = 1 a 3 [ 1 2 ( a x + b ) 2 − 2 b ( a x + b ) + b 2 ln ⁡ ∣ a x + b ∣ ] + C 4.\int{\frac{x^2}{ax+b}dx}=\frac{1}{a^3}[\frac{1}{2}(ax+b)^2-2b(ax+b)+b^2\ln{\mid ax+b \mid}]+C 4.ax+bx2dx=a31[21(ax+b)22b(ax+b)+b2lnax+b]+C

    5. ∫ d x x ( a x + b ) = − 1 b ln ⁡ ∣ a x + b x ∣ + C 5.\int{\frac{dx}{x(ax+b)}}=-\frac{1}{b}\ln{\mid \frac{ax+b}{x} \mid}+C 5.x(ax+b)dx=b1lnxax+b+C

    6. ∫ d x x 2 ( a x + b ) = − 1 b x + a b 2 ln ⁡ ∣ a x + b x ∣ + C 6.\int{\frac{dx}{x^2(ax+b)}}=-\frac{1}{bx}+\frac{a}{b^2}\ln{\mid \frac{ax+b}{x} \mid}+C 6.x2(ax+b)dx=bx1+b2alnxax+b+C

    7. ∫ x ( a x + b ) 2 d x = 1 a 2 ( ln ⁡ ∣ a x + b ∣ + b a x + b ) + C 7.\int{\frac{x}{(ax+b)^2}dx}=\frac{1}{a^2}(\ln{\mid ax+b \mid}+\frac{b}{ax+b})+C 7.(ax+b)2xdx=a21(lnax+b+ax+bb)+C

    8. ∫ x 2 ( a x + b ) 2 d x = 1 a 3 ( a x + b − 2 b ln ⁡ ∣ a x + b ∣ − b 2 a x + b ) + C 8.\int{\frac{x^2}{(ax+b)^2}dx}=\frac{1}{a^3}(ax+b-2b\ln{\mid ax+b \mid-\frac{b^2}{ax+b}})+C 8.(ax+b)2x2dx=a31(ax+b2blnax+bax+bb2)+C

    9. ∫ d x x ( a x + b ) 2 = 1 b ( a x + b ) − 1 b 2 ln ⁡ ∣ a x + b x ∣ + C 9.\int{\frac{dx}{x(ax+b)^2}}=\frac{1}{b(ax+b)}-\frac{1}{b^2}\ln{\mid \frac{ax+b}{x} \mid}+C 9.x(ax+b)2dx=b(ax+b)1b21lnxax+b+C

    10. ∫ d x x 2 ( a x + b ) 2 = − 1 b 2 [ 2 a x + b x ( a x + b ) + 2 a b ln ⁡ ∣ x a x + b ∣ ] + C 10.\int\frac{dx}{x^2(ax+b)^2}=-\frac{1}{b^2}[\frac{2ax+b}{x(ax+b)}+\frac{2a}{b}\ln|\frac{x}{ax+b}|]+C 10.x2(ax+b)2dx=b21[x(ax+b)2ax+b+b2alnax+bx]+C

    11. ∫ x 2 ( a x + b ) 3 d x = 1 a 3 [ 2 b a x + b − b 2 2 ( a x + b ) 2 + ln ⁡ ∣ a x + b ∣ ] + C 11.\int\frac{x^2}{(ax+b)^3}dx=\frac{1}{a^3}[\frac{2b}{ax+b}-\frac{b^2}{2(ax+b)^2}+\ln|ax+b|]+C 11.(ax+b)3x2dx=a31[ax+b2b2(ax+b)2b2+lnax+b]+C

    12. ∫ x ( a x + b ) n d x = 1 a 2 [ − 1 ( n − 2 ) ( a x + b ) n − 2 + b ( n − 1 ) ( a x + b ) n − 1 ] + C   ( n ≠ 1 , 2 ) 12.\int\frac{x}{(ax+b)^n}dx=\frac{1}{a^2}[-\frac{1}{(n-2)(ax+b)^{n-2}}+\frac{b}{(n-1)(ax+b)^{n-1}}]+C\,(n≠1,2) 12.(ax+b)nxdx=a21[(n2)(ax+b)n21+(n1)(ax+b)n1b]+C(n=1,2)

    13. ∫ x 2 ( a x + b ) n d x = 1 a 3 [ − 1 ( n − 3 ) ( a x + b ) n − 3 + 2 b ( n − 2 ) ( a x + b ) n − 2 − b 2 ( n − 1 ) ( a x + b ) n − 1 ] + C   ( n ≠ 1 , 2 , 3 ) 13.\int\frac{x^2}{(ax+b)^n}dx=\frac{1}{a^3}[-\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}}]+C\,(n≠1,2,3) 13.(ax+b)nx2dx=a31[(n3)(ax+b)n31+(n2)(ax+b)n22b(n1)(ax+b)n1b2]+C(n=1,2,3)

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    三.含有 a x 2 + b   ( a > 0 ) ax^2+b\,(a>0) ax2+b(a>0)的形式

    1. ∫ d x a x 2 + b d x = { 1 a b arctan ⁡ a b x + C ( b > 0 ) 1 2 − a b ln ⁡ ∣ x − a x + a ∣ + C ( b < 0 ) 1.\int{\frac{dx}{ax^2+b}dx}=\begin{cases}\frac{1}{\sqrt{ab}}\arctan{\sqrt{\frac{a}{b}}x}+C(b>0)\\\frac{1}{2{\sqrt{-ab}}}\ln{\mid\frac{x-a}{x+a} \mid}+C(b<0) \end{cases} 1.ax2+bdxdx={ab 1arctanba x+C(b>0)2ab 1lnx+axa+C(b<0)

    2. ∫ x a x 2 + b d x = 1 2 a ln ⁡ ∣ a x 2 + b ∣ + C 2.\int\frac{x}{ax^2+b}dx=\frac{1}{2a}\ln|ax^2+b|+C 2.ax2+bxdx=2a1lnax2+b+C

    3. ∫ x 2 a x 2 + b d x = x a − b a ∫ d x a x 2 + b 3.\int\frac{x^2}{ax^2+b}dx=\frac{x}{a}-\frac{b}{a}\int\frac{dx}{ax^2+b} 3.ax2+bx2dx=axabax2+bdx

    4. ∫ d x x ( a x 2 + b ) = 1 2 b ln ⁡ x 2 ∣ a x 2 + b ∣ + C 4.\int\frac{dx}{x(ax^2+b)}=\frac{1}{2b}\ln\frac{x^2}{|ax^2+b|}+C 4.x(ax2+b)dx=2b1lnax2+bx2+C

    5. ∫ d x x 2 ( a x 2 + b ) = − 1 b x − a b ∫ d x a x 2 + b 5.\int\frac{dx}{x^2(ax^2+b)}=-\frac{1}{bx}-\frac{a}{b}\int\frac{dx}{ax^2+b} 5.x2(ax2+b)dx=bx1baax2+bdx

    6. ∫ d x x 3 ( a x 2 + b ) = a 2 b 2 ln ⁡ ∣ a x 2 + b ∣ x 2 − 1 2 b x 2 + C 6.\int\frac{dx}{x^3(ax^2+b)}=\frac{a}{2b^2}\ln\frac{|ax^2+b|}{x^2}-\frac{1}{2bx^2}+C 6.x3(ax2+b)dx=2b2alnx2ax2+b2bx21+C

    7. ∫ d x ( a x 2 + b ) 2 = x 2 b ( a x 2 + b ) + 1 2 b ∫ d x a x 2 + b 7.\int\frac{dx}{(ax^2+b)^2}=\frac{x}{2b(ax^2+b)}+\frac{1}{2b}\int\frac{dx}{ax^2+b} 7.(ax2+b)2dx=2b(ax2+b)x+2b1ax2+bdx

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    四.含有 ± x 2 ± a 2   ( a > 0 ) ±x^2±a^2\,(a>0) ±x2±a2(a>0)的形式
    1. ∫ d x x 2 + a 2 = 1 a arctan ⁡ x a + C 1.\int{\frac{dx}{x^2+a^2}}=\frac{1}{a}\arctan{\frac{x}{a}}+C 1.x2+a2dx=a1arctanax+C

    2. ∫ d x ( a 2 ± x 2 ) n = 1 2 a 2 ( n − 1 ) [ x ( a 2 ± x 2 ) n − 1 + ( 2 n − 3 ) ∫ d x ( a 2 ± x 2 ) n − 1 ]   ( n ≠ 1 ) 2.\int{\frac{dx}{(a^2±x^2)^n}}=\frac{1}{2a^2(n-1)}[\frac{x}{(a^2±x^2)^{n-1}}+(2n-3)\int{\frac{dx}{(a^2±x^2)^{n-1}}}]\,(n≠1) 2.(a2±x2)ndx=2a2(n1)1[(a2±x2)n1x+(2n3)(a2±x2)n1dx](n=1)

    3. ∫ d x x 2 − a 2 = 1 2 a ln ⁡ ∣ x − a x + a ∣ + C 3.\int{\frac{dx}{x^2-a^2}}=\frac{1}{2a}\ln{\mid \frac{x-a}{x+a} \mid}+C 3.x2a2dx=2a1lnx+axa+C

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    五.含有 a x 2 + b x + c ax^2+bx+c ax2+bx+c的形式
    1. ∫ d x a x 2 + b x + c = { 2 4 a c − b 2 a r c t a n 2 a x + b 4 a c − b 2 + C   ( b 2 < 4 a c ) 1 b 2 − 4 a c l n ∣ 2 a x + b − b 2 − 4 a c 2 a x + b + b 2 − 4 a c ∣ + C   ( b 2 > 4 a c ) 1.\int \frac{dx}{ax^2+bx+c}=\begin{cases}\frac{2}{\sqrt{4ac-b^2}}arctan\frac{2ax+b}{\sqrt{4ac-b^2}}+C\,(b^2<4ac)\\ \frac{1}{b^2-4ac}ln|\frac{2ax+b-\sqrt{b^2-4ac}}{2ax+b+\sqrt{b^2-4ac}}|+C\,(b^2>4ac)\end{cases} 1.ax2+bx+cdx={4acb2 2arctan4acb2 2ax+b+C(b2<4ac)b24ac1ln2ax+b+b24ac 2ax+bb24ac +C(b2>4ac)

    2. ∫ x a x 2 + b x + c d x = 1 2 a l n ∣ a x 2 + b x + c ∣ − b 2 a ∫ d x a x 2 + b x + c 2.\int \frac{x}{ax^2+bx+c}dx=\frac{1}{2a}ln|ax^2+bx+c|-\frac{b}{2a}\int \frac{dx}{ax^2+bx+c} 2.ax2+bx+cxdx=2a1lnax2+bx+c2abax2+bx+cdx

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    六.含有 a x + b \sqrt{ax+b} ax+b 的形式

    1. ∫ a x + b d x 1.\int{\sqrt{ax+b}dx} 1.ax+b dx = 2 3 a ( a x + b ) 3 + C \frac{2}{3a}\sqrt{(ax+b)^3}+C 3a2(ax+b)3 +C

    2. ∫ x a x + b d x 2.\int{x\sqrt{ax+b}dx} 2.xax+b dx = 2 15 a 2 ( 3 a x − 2 b ) ( a x + b ) 3 + C \frac{2}{15a^2}(3ax-2b)\sqrt{(ax+b)^3}+C 15a22(3ax2b)(ax+b)3 +C

    3. ∫ x 2 a x + b d x 3.\int{x^2\sqrt{ax+b}dx} 3.x2ax+b dx = 2 15 a 2 ( 15 a 2 x 2 − 12 a b x + 8 b 2 ) ( a x + b ) 3 + C \frac{2}{15a^2}(15a^2x^2-12abx+8b^2)\sqrt{(ax+b)^3}+C 15a22(15a2x212abx+8b2)(ax+b)3 +C

    4. ∫ x n a x + b d x = 2 a ( 2 n + 3 ) [ x n ( a x + b ) 3 2 − n b ∫ x n − 1 a x + b d x ] 4.\int{x^n\sqrt{ax+b}}dx=\frac{2}{a(2n+3)}[x^n(ax+b)^{\frac{3}{2}}-nb\int x^{n-1}\sqrt{ax+b}dx] 4.xnax+b dx=a(2n+3)2[xn(ax+b)23nbxn1ax+b dx]

    5. ∫ x a x + b d x = 2 3 a 2 ( a x − 2 b ) a x + b + C 5.\int{\frac{x}{\sqrt{ax+b}}dx}=\frac{2}{3a^2}(ax-2b)\sqrt{ax+b}+C 5.ax+b xdx=3a22(ax2b)ax+b +C

    6. ∫ x 2 a x + b d x = 2 15 a 3 ( 3 a 2 x 2 − 4 a b c x + 8 b 2 ) a x + b + C 6.\int{\frac{x^2}{\sqrt{ax+b}}dx}=\frac{2}{15a^3}(3a^2x^2-4abcx+8b^2)\sqrt{ax+b}+C 6.ax+b x2dx=15a32(3a2x24abcx+8b2)ax+b +C

    7. ∫ x n a x + b d x = 2 ( 2 n + 1 ) a ( x n a x + b − n b ∫ x n − 1 a x + b d x ) 7.\int\frac{x^n}{\sqrt{ax+b}}dx=\frac{2}{(2n+1)a}(x^n\sqrt{ax+b}-nb\int\frac{x^{n-1}}{\sqrt{ax+b}}dx) 7.ax+b xndx=(2n+1)a2(xnax+b nbax+b xn1dx)

    8. ∫ d x x a x + b = { 1 b ln ⁡ ∣ a x + b − b a x + b + b ∣ + C ( b > 0 ) 2 − b arctan ⁡ a x + b − b + C ( b < 0 ) 8.\int{\frac{dx}{x\sqrt{ax+b}}}=\begin{cases} \frac{1}{\sqrt{b}}\ln{\mid \frac{\sqrt{ax+b}-\sqrt{b}}{\sqrt{ax+b}+\sqrt{b}} \mid}+C(b>0)\\\frac{2}{\sqrt{-b}}\arctan{\sqrt{\frac{ax+b}{-b}}}+C(b<0) \end{cases} 8.xax+b dx=b 1lnax+b +b ax+b b +C(b>0)b 2arctanbax+b +C(b<0)

    9. ∫ d x x 2 a x + b d x = − a x + b b x − a 2 b ∫ d x x a x + b 9.\int{\frac{dx}{x^2\sqrt{ax+b}}dx}=-\frac{\sqrt{ax+b}}{bx}-\frac{a}{2b}\int{\frac{dx}{x\sqrt{ax+b}}} 9.x2ax+b dxdx=bxax+b 2baxax+b dx

    10. ∫ d x x n a x + b = − 1 b ( n − 1 ) [ a x + b x n − 1 + a ( 2 n − 3 ) 2 ∫ d x x n − 1 a x + b ]   ( n ≠ − 1 ) 10.\int\frac{dx}{x^n\sqrt{ax+b}}=-\frac{1}{b(n-1)}[\frac{\sqrt{ax+b}}{x^{n-1}}+\frac{a(2n-3)}{2}\int\frac{dx}{x^{n-1}\sqrt{ax+b}}]\,(n≠-1) 10.xnax+b dx=b(n1)1[xn1ax+b +2a(2n3)xn1ax+b dx](n=1)

    11. ∫ a x + b x d x = 2 a x + b + b ∫ d x x a x + b 11.\int{\frac{\sqrt{ax+b}}{x}dx}=2\sqrt{ax+b}+b\int{\frac{dx}{x\sqrt{ax+b}}} 11.xax+b dx=2ax+b +bxax+b dx

    12. ∫ a x + b x 2 d x = − a x + b x + a 2 ∫ d x x a x + b 12.\int{\frac{\sqrt{ax+b}}{x^2}dx}=-\frac{\sqrt{ax+b}}{x}+\frac{a}{2}\int{\frac{dx}{x\sqrt{ax+b}}} 12.x2ax+b dx=xax+b +2axax+b dx

    13. ∫ a x + b x n d x = − 1 b ( n − 1 ) [ ( a x + b ) 3 2 x n − 1 + ( 2 n − 5 ) a 2 ∫ a x + b x n − 1 d x ]   ( n ≠ − 1 ) 13.\int{\frac{\sqrt{ax+b}}{x^n}dx}=-\frac{1}{b(n-1)}[\frac{(ax+b)^{\frac{3}{2}}}{x^{n-1}}+\frac{(2n-5)a}{2}\int{\frac{\sqrt{ax+b}}{x^{n-1}}dx}]\,(n≠-1) 13.xnax+b dx=b(n1)1[xn1(ax+b)23+2(2n5)axn1ax+b dx](n=1)

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    七.含有 ± a x 2 + b x + c   ( a > 0 ) \sqrt{±ax^2+bx+c}\,(a>0) ±ax2+bx+c (a>0)的形式

    1. ∫ d x a x 2 + b x + c = 1 a ln ⁡ ∣ 2 a x + b + 2 a a x 2 + b x + c ∣ + C 1.\int\frac{dx}{\sqrt{ax^2+bx+c}}=\frac{1}{\sqrt{a}}\ln|2ax+b+2\sqrt{a}\sqrt{ax^2+bx+c}|+C 1.ax2+bx+c dx=a 1ln2ax+b+2a ax2+bx+c +C

    2. ∫ a x 2 + b x + c d x = 2 a x + b 4 a a x 2 + b x + c + 4 a c − b 2 8 a 3 2 ln ⁡ ∣ 2 a x + b + 2 a a x 2 + b x + c ∣ + C 2.\int\sqrt{ax^2+bx+c}dx=\frac{2ax+b}{4a}\sqrt{ax^2+bx+c}+\frac{4ac-b^2}{8a^{\frac{3}{2}}}\ln|2ax+b+2\sqrt{a}\sqrt{ax^2+bx+c}|+C 2.ax2+bx+c dx=4a2ax+bax2+bx+c +8a234acb2ln2ax+b+2a ax2+bx+c +C

    3. ∫ x a x 2 + b x + c d x = 1 a a x 2 + b x + c − b 2 a 3 2 ln ⁡ ∣ 2 a x + b + 2 a a x 2 + b x + c ∣ + C 3.\int\frac{x}{\sqrt{ax^2+bx+c}}dx=\frac{1}{a}\sqrt{ax^2+bx+c}-\frac{b}{2a^{\frac{3}{2}}}\ln|2ax+b+2\sqrt{a}\sqrt{ax^2+bx+c}|+C 3.ax2+bx+c xdx=a1ax2+bx+c 2a23bln2ax+b+2a ax2+bx+c +C

    4. ∫ d x − a x 2 + b x + c = 1 a arcsin ⁡ 2 a x − b b 2 + 4 a c + C 4.\int\frac{dx}{\sqrt{-ax^2+bx+c}}=\frac{1}{\sqrt{a}}\arcsin\frac{2ax-b}{\sqrt{b^2+4ac}}+C 4.ax2+bx+c dx=a 1arcsinb2+4ac 2axb+C

    5. ∫ − a x 2 + b x + c d x = 2 a x − b 4 a − a x 2 + b x + c + 4 a c + b 2 8 a 3 2 arcsin ⁡ 2 a x − b b 2 + 4 a c + C 5.\int\sqrt{-ax^2+bx+c}dx=\frac{2ax-b}{4a}\sqrt{-ax^2+bx+c}+\frac{4ac+b^2}{8a^{\frac{3}{2}}}\arcsin\frac{2ax-b}{\sqrt{b^2+4ac}}+C 5.ax2+bx+c dx=4a2axbax2+bx+c +8a234ac+b2arcsinb2+4ac 2axb+C

    6. ∫ x − a x 2 + b x + c d x = − 1 a − a x 2 + b x + c + b 2 a 3 2 arcsin ⁡ 2 a x − b b 2 + 4 a c + C 6.\int\frac{x}{\sqrt{-ax^2+bx+c}}dx=-\frac{1}{a}\sqrt{-ax^2+bx+c}+\frac{b}{2a^{\frac{3}{2}}}\arcsin\frac{2ax-b}{\sqrt{b^2+4ac}}+C 6.ax2+bx+c xdx=a1ax2+bx+c +2a23barcsinb2+4ac 2axb+C

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    八.含有 x 2 ± a 2 ( a > 0 ) \sqrt{x^2±a^2}(a>0) x2±a2 (a>0)的形式

    1. ∫ x 2 ± a 2 d x = 1 2 ( x x 2 ± a 2 ± a 2 l n   ∣ x + x 2 ± a 2 ∣ ) + C 1.\int\sqrt{x^2±a^2}dx=\frac{1}{2}(x\sqrt{x^2±a^2}±a^2ln\,|x+\sqrt{x^2±a^2}|)+C 1.x2±a2 dx=21(xx2±a2 ±a2lnx+x2±a2 )+C

    2. ∫ ( x 2 + a 2 ) 3 2 = x 8 ( 2 x 2 + 5 a 2 ) x 2 + a 2 + 3 8 a 4 ln ⁡ ( x + x 2 + a 2 ) + C 2.\int(x^2+a^2)^{\frac{3}{2}}=\frac{x}{8}(2x^2+5a^2)\sqrt{x^2+a^2}+\frac{3}{8}a^4\ln(x+\sqrt{x^2+a^2})+C 2.(x2+a2)23=8x(2x2+5a2)x2+a2 +83a4ln(x+x2+a2 )+C

    3. ∫ ( x 2 − a 2 ) 3 2 = x 8 ( 2 x 2 − 5 a 2 ) x 2 − a 2 + 3 8 a 4 ln ⁡ ∣ x + x 2 − a 2 ∣ + C 3.\int(x^2-a^2)^{\frac{3}{2}}=\frac{x}{8}(2x^2-5a^2)\sqrt{x^2-a^2}+\frac{3}{8}a^4\ln|x+\sqrt{x^2-a^2}|+C 3.(x2a2)23=8x(2x25a2)x2a2 +83a4lnx+x2a2 +C

    4. ∫ x x 2 ± a 2 d x = 1 3 ( x 2 ± a 2 ) 3 2 + C 4.\int x\sqrt{x^2±a^2}dx=\frac{1}{3}(x^2±a^2)^{\frac{3}{2}}+C 4.xx2±a2 dx=31(x2±a2)23+C

    5. ∫ x 2 x 2 ± a 2 d x = 1 8 [ x ( 2 x 2 ± a 2 ) x 2 ± a 2 − a 4 l n   ∣ x + x 2 ± a 2 ∣ ] + C 5.\int x^2\sqrt{x^2±a^2}dx=\frac{1}{8}[x(2x^2±a^2)\sqrt{x^2±a^2}-a^4ln\,|x+\sqrt{x^2±a^2}|]+C 5.x2x2±a2 dx=81[x(2x2±a2)x2±a2 a4lnx+x2±a2 ]+C

    6. ∫ 1 x x 2 + a 2 d x = x 2 + a 2 − a l n   ∣ a + x 2 + a 2 x ∣ + C 6.\int\frac{1}{x}\sqrt{x^2+a^2}dx=\sqrt{x^2+a^2}-aln\,|\frac{a+\sqrt{x^2+a^2}}{x}|+C 6.x1x2+a2 dx=x2+a2 alnxa+x2+a2 +C

    7. ∫ 1 x x 2 − a 2 d x = x 2 − a 2 − a arccos ⁡ a x + C 7.\int\frac{1}{x}\sqrt{x^2-a^2}dx=\sqrt{x^2-a^2}-a\arccos\frac{a}{x}+C 7.x1x2a2 dx=x2a2 aarccosxa+C

    8. ∫ 1 x 2 x 2 ± a 2 d x = − 1 x x 2 ± a 2 + l n   ∣ x + x 2 ± a 2 ∣ + C 8.\int\frac{1}{x^2}\sqrt{x^2±a^2}dx=-\frac{1}{x}\sqrt{x^2±a^2}+ln\,|x+\sqrt{x^2±a^2}|+C 8.x21x2±a2 dx=x1x2±a2 +lnx+x2±a2 +C

    9. ∫ d x x 2 + a 2 = a r s h x a + C = l n   ∣ x + x 2 + a 2 ∣ + C 9.\int\frac{dx}{\sqrt{x^2+a^2}}=arsh\frac{x}{a}+C=ln\,|x+\sqrt{x^2+a^2}|+C 9.x2+a2 dx=arshax+C=lnx+x2+a2 +C

    10. ∫ d x x 2 − a 2 = x ∣ x ∣ a r s h ∣ x ∣ a + C = l n   ∣ x + x 2 − a 2 ∣ + C 10.\int\frac{dx}{\sqrt{x^2-a^2}}=\frac{x}{|x|}arsh\frac{|x|}{a}+C=ln\,|x+\sqrt{x^2-a^2}|+C 10.x2a2 dx=xxarshax+C=lnx+x2a2 +C

    11. ∫ x x 2 ± a 2 d x = x 2 ± a 2 + C 11.\int\frac{x}{\sqrt{x^2±a^2}}dx=\sqrt{x^2±a^2}+C 11.x2±a2 xdx=x2±a2 +C

    12. ∫ x 2 x 2 ± a 2 d x = 1 2 ( x x 2 ± a 2 ∓ a 2 l n   ∣ x + x 2 ± a 2 ∣ ) + C 12.\int\frac{x^2}{\sqrt{x^2±a^2}}dx=\frac{1}{2}(x\sqrt{x^2±a^2}∓a^2ln\,|x+\sqrt{x^2±a^2}|)+C 12.x2±a2 x2dx=21(xx2±a2 a2lnx+x2±a2 )+C

    13. ∫ d x x x 2 + a 2 = − 1 a l n   ∣ a + x 2 + a 2 x ∣ + C 13.\int\frac{dx}{x\sqrt{x^2+a^2}}=-\frac{1}{a}ln\,|\frac{a+\sqrt{x^2+a^2}}{x}|+C 13.xx2+a2 dx=a1lnxa+x2+a2 +C

    14. ∫ d x x x 2 − a 2 = 1 a arccos ⁡ a x + C 14.\int\frac{dx}{x\sqrt{x^2-a^2}}=\frac{1}{a}\arccos\frac{a}{x}+C 14.xx2a2 dx=a1arccosxa+C

    15. ∫ d x x 2 x 2 ± a 2 = ∓ x 2 ± a 2 a 2 x + C 15.\int\frac{dx}{x^2\sqrt{x^2±a^2}}=∓\frac{\sqrt{x^2±a^2}}{a^2x}+C 15.x2x2±a2 dx=a2xx2±a2 +C

    16. ∫ d x ( x 2 ± a 2 ) 3 2 = ± x a 2 x 2 ± a 2 + C 16.\int\frac{dx}{(x^2±a^2)^{\frac{3}{2}}}=\frac{±x}{a^2\sqrt{x^2±a^2}}+C 16.(x2±a2)23dx=a2x2±a2 ±x+C

    17. ∫ x ( x 2 ± a 2 ) 3 2 d x = − 1 x 2 ± a 2 + C 17.\int\frac{x}{(x^2±a^2)^{\frac{3}{2}}}dx=-\frac{1}{\sqrt{x^2±a^2}}+C 17.(x2±a2)23xdx=x2±a2 1+C

    18. ∫ x 2 ( x 2 ± a 2 ) 3 2 d x = − x x 2 ± a 2 + ln ⁡ ( x + x 2 ± a 2 ) + C 18.\int\frac{x^2}{(x^2±a^2)^{\frac{3}{2}}}dx=-\frac{x}{\sqrt{x^2±a^2}}+\ln(x+\sqrt{x^2±a^2})+C 18.(x2±a2)23x2dx=x2±a2 x+ln(x+x2±a2 )+C

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    九.含有 a 2 − x 2 ( a > 0 ) \sqrt{a^2-x^2}(a>0) a2x2 (a>0)的形式*

    1. ∫ a 2 − x 2 d x = 1 2 ( x a 2 − x 2 + a 2 arcsin ⁡ x a ) + C 1.\int\sqrt{a^2-x^2}dx=\frac{1}{2}(x\sqrt{a^2-x^2}+a^2\arcsin\frac{x}{a})+C 1.a2x2 dx=21(xa2x2 +a2arcsinax)+C

    2. ∫ ( a 2 − x 2 ) 3 2 d x = 5 8 ( 5 a 2 − 2 x 2 ) a 2 − x 2 + 3 8 a 4 arcsin ⁡ x a + C 2.\int(a^2-x^2)^{\frac{3}{2}}dx=\frac{5}{8}(5a^2-2x^2)\sqrt{a^2-x^2}+\frac{3}{8}a^4\arcsin\frac{x}{a}+C 2.(a2x2)23dx=85(5a22x2)a2x2 +83a4arcsinax+C

    3. ∫ x a 2 − x 2 d x = − 1 3 ( a 2 − x 2 ) 3 2 + C 3.\int x\sqrt{a^2-x^2}dx=-\frac{1}{3}(a^2-x^2)^{\frac{3}{2}}+C 3.xa2x2 dx=31(a2x2)23+C

    4. ∫ x 2 a 2 − x 2 d x = 1 8 [ x ( 2 x 2 − a 2 ) a 2 − x 2 + a 4 arcsin ⁡ x a ] + C 4.\int x^2\sqrt{a^2-x^2}dx=\frac{1}{8}[x(2x^2-a^2)\sqrt{a^2-x^2}+a^4\arcsin\frac{x}{a}]+C 4.x2a2x2 dx=81[x(2x2a2)a2x2 +a4arcsinax]+C

    5. ∫ 1 x a 2 − x 2 d x = a 2 − x 2 − a ln ⁡ ∣ a + a 2 − x 2 x ∣ + C 5.\int\frac{1}{x}\sqrt{a^2-x^2}dx=\sqrt{a^2-x^2}-a\ln|\frac{a+\sqrt{a^2-x^2}}{x}|+C 5.x1a2x2 dx=a2x2 alnxa+a2x2 +C

    6. ∫ 1 x 2 a 2 − x 2 d x = − 1 x a 2 − x 2 − arcsin ⁡ x a + C 6.\int\frac{1}{x^2}\sqrt{a^2-x^2}dx=-\frac{1}{x}\sqrt{a^2-x^2}-\arcsin\frac{x}{a}+C 6.x21a2x2 dx=x1a2x2 arcsinax+C

    7. ∫ d x a 2 − x 2 arcsin ⁡ x a + C 7.\int\frac{dx}{\sqrt{a^2-x^2}}\arcsin\frac{x}{a}+C 7.a2x2 dxarcsinax+C

    8. ∫ d x x a 2 − x 2 = − 1 a ln ⁡ ∣ a + a 2 − x 2 x ∣ + C 8.\int\frac{dx}{x\sqrt{a^2-x^2}}=-\frac{1}{a}\ln|\frac{a+\sqrt{a^2-x^2}}{x}|+C 8.xa2x2 dx=a1lnxa+a2x2 +C

    9. ∫ d x x 2 a 2 − x 2 = − a 2 − x 2 a 2 x + C 9.\int\frac{dx}{x^2\sqrt{a^2-x^2}}=-\frac{\sqrt{a^2-x^2}}{a^2x}+C 9.x2a2x2 dx=a2xa2x2 +C

    10. ∫ x a 2 − x 2 d x = − a 2 − x 2 + C 10.\int\frac{x}{\sqrt{a^2-x^2}}dx=-\sqrt{a^2-x^2}+C 10.a2x2 xdx=a2x2 +C

    11. ∫ x 2 a 2 − x 2 d x = 1 2 ( − x a 2 − x 2 + a 2 arcsin ⁡ x a ) + C 11.\int\frac{x^2}{\sqrt{a^2-x^2}}dx=\frac{1}{2}(-x\sqrt{a^2-x^2}+a^2\arcsin\frac{x}{a})+C 11.a2x2 x2dx=21(xa2x2 +a2arcsinax)+C

    12. ∫ d x ( a 2 − x 2 ) 3 2 = x a 2 a 2 − x 2 + C 12.\int\frac{dx}{(a^2-x^2)^{\frac{3}{2}}}=\frac{x}{a^2\sqrt{a^2-x^2}}+C 12.(a2x2)23dx=a2a2x2 x+C

    13. ∫ x ( a 2 − x 2 ) 3 2 d x = 1 a 2 − x 2 + C 13.\int\frac{x}{(a^2-x^2)^{\frac{3}{2}}}dx=\frac{1}{\sqrt{a^2-x^2}}+C 13.(a2x2)23xdx=a2x2 1+C

    14. ∫ x 2 ( a 2 − x 2 ) 3 2 d x = x a 2 − x 2 − arcsin ⁡ x a + C 14.\int\frac{x^2}{(a^2-x^2)^{\frac{3}{2}}}dx=\frac{x}{\sqrt{a^2-x^2}}-\arcsin\frac{x}{a}+C 14.(a2x2)23x2dx=a2x2 xarcsinax+C

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    十.含有 ± x − a x − b \sqrt{±\frac{x-a}{x-b}} ±xbxa ( x − a ) ( b − x ) \sqrt{(x-a)(b-x)} (xa)(bx) 的形式

    1. ∫ x − a x − b = ( x − b ) x − a x − b + ( b − a ) ln ⁡ ( ∣ x − a ∣ + ∣ x − b ∣ ) + C 1.\int\sqrt{\frac{x-a}{x-b}}=(x-b)\sqrt{\frac{x-a}{x-b}}+(b-a)\ln(\sqrt{|x-a|}+\sqrt{|x-b|})+C 1.