• 积分常用公式表
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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 ( x μ ) ′ = μ x μ − 1 (x^\mu)'=\mu x^{\mu-1} ∫ x μ d x = x μ + 1 μ + 1 + C \int x^\mu \mathbf{d}x=\cfrac{x^{\mu+1}}{\mu+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 ( a x ) ′ = a x ln ⁡ a (a^x)'=a^x \ln a ∫ a x d x = a x ln ⁡ a + C \int a^x\mathbf{d}x=\cfrac{a^x}{\ln a}+C
3 d ( e x ) = e x   d x \mathbf{d}(e^x)=e^x\ \mathbf{d}x ( e x ) ′ = e x (e^x)'=e^x ∫ e x d x = e x + C \int e^x \mathbf{d}x=e^x+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 ( log ⁡ a x ) ′ = 1 x ln ⁡ a (\log_ax)'=\cfrac{1}{x\ln a}
5 d ( ln ⁡ x ) = 1 x d x \mathbf{d}(\ln x)=\cfrac{1}{x}\mathbf{d}x ( ln ⁡ x ) ′ = 1 x (\ln x)'=\cfrac{1}{x} ∫ 1 x d x = ln ⁡ ∣ x ∣ + C \int \cfrac{1}{x} \mathbf{d}x=\ln \vert x \vert+C
|三角函数
6 d ( sin ⁡ x ) = cos ⁡ x   d x \mathbf{d}(\sin x)=\cos x\ \mathbf{d}x ( sin ⁡ x ) ′ = cos ⁡ x (\sin x)'=\cos x ∫ cos ⁡ x d x = sin ⁡ x + C \int \cos x\mathbf{d}x=\sin x+C
7 d ( cos ⁡ x ) = − sin ⁡ x   d x \mathbf{d}(\cos x)=-\sin x\ \mathbf{d}x ( cos ⁡ x ) ′ = − sin ⁡ x (\cos x)'=-\sin x ∫ sin ⁡ x d x = − cos ⁡ x + C \int \sin x \mathbf{d}x=-\cos x+C
8 d ( tan ⁡ x ) = sec ⁡ 2 x   d x \mathbf{d}(\tan x)=\sec^2x\ \mathbf{d}x ( tan ⁡ x ) ′ = sec ⁡ 2 x (\tan x)'=\sec^2x ∫ 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
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9 d ( cot ⁡ x ) = − csc ⁡ 2 x   d x \mathbf{d}(\cot x)=-\csc^2x\ \mathbf{d}x ( cot ⁡ x ) ′ = − csc ⁡ 2 x (\cot x)'=-\csc^2x ∫ 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
10 d ( sec ⁡ x ) = sec ⁡ x   tan ⁡ x   d x \mathbf{d}(\sec x)=\sec x\ \tan x\ \mathbf{d}x ( sec ⁡ x ) ′ = sec ⁡ x tan ⁡ x (\sec x)'=\sec x\tan x ∫ sec ⁡ x tan ⁡ x d x = sec ⁡ x + C \int \sec x \tan x\mathbf{d}x=\sec x+C
11 d ( csc ⁡ x ) = − csc ⁡ x cot ⁡ x   d x \mathbf{d}(\csc x)=-\csc x\cot x\ \mathbf{d}x ( csc ⁡ x ) ′ = − csc ⁡ x cot ⁡ x (\csc x)'=-\csc x\cot x ∫ csc ⁡ x cot ⁡ x d x = − csc ⁡ x + C \int \csc x \cot x \mathbf{d}x=-\csc x+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 ( arcsin ⁡ x ) ′ = 1 1 − x 2 (\arcsin x)'=\cfrac{1}{\sqrt{1-x^2}} ∫ 1 1 − x 2 = arcsin ⁡ x + C \int \cfrac{1}{\sqrt{1-x^2}}=\arcsin x+C
13 d ( arccos ⁡ x ) = − 1 1 − x 2 d x \mathbf{d}(\arccos x)=-\cfrac{1}{\sqrt{1-x^2}}\mathbf{d}x ( arccos ⁡ x ) ′ = − 1 1 − x 2 (\arccos x)'=-\cfrac{1}{\sqrt{1-x^2}}
14 d ( arctan ⁡ x ) = 1 1 + x 2 d x \mathbf{d}(\arctan x)=\cfrac{1}{1+x^2}\mathbf{d}x ( arctan ⁡ x ) ′ = 1 1 + x 2 (\arctan x)'=\cfrac{1}{1+x^2} ∫ 1 1 + x 2 = arctan ⁡ x + C \int \cfrac{1}{1+x^2}=\arctan x+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 ( a r c c o t x ) ′ = − 1 1 + x 2 (arccot x)'=-\cfrac{1}{1+x^2}

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）

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

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

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

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

∫ 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

∫ 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

∫ 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

∫ 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

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十三、含有指数函数的积分

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• 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 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)

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

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二.含有 a x + 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

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)

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

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

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

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

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

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

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

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

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

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)

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)

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三.含有 a x 2 + b   ( a > 0 ) ax^2+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}

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

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}

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

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}

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

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}

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四.含有 ± x 2 ± a 2   ( a > 0 ) ±x^2±a^2\,(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

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)

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

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五.含有 a x 2 + b x + c ax^2+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}

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}

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六.含有 a x + b \sqrt{ax+b} 的形式

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

2. ∫ x a x + b d x 2.\int{x\sqrt{ax+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

3. ∫ x 2 a x + b d x 3.\int{x^2\sqrt{ax+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

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]

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

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

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)

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}

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}}}

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)

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}}}

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}}}

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)

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七.含有 ± a x 2 + b x + c   ( a > 0 ) \sqrt{±ax^2+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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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十.含有 ± x − a x − b \sqrt{±\frac{x-a}{x-b}} ( x − a ) ( b − x ) \sqrt{(x-a)(b-x)} 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