最全常用高数公式

文章目录

  • 1 等价无穷小 ( x → 0 x\to 0 x0)
  • 2 常用公式
    • 2.1 和式夹逼准则的两个思路
    • 2.2 小基础
    • 2.3 一元二次方程解
  • 3 ★ \bigstar 常用展开公式 ★ \bigstar
  • 4 常用不等式
  • 5 三角变换
  • 6 ★ \bigstar 微分 ★ \bigstar
    • 6.1 定义式
    • 6.2 常用难记微分公式
    • 6.3 分析函数关注点
  • 7 ★ \bigstar 积分 ★ \bigstar
    • 7.1 定积分定义
    • 7.2 基本积分表
    • 7.3 常用积分公式
    • 7.4 积分常用方法

1 等价无穷小 ( x → 0 x\to 0 x0)

s i n x ∼ x sinx\sim x sinxx, t a n x ∼ x tanx\sim x tanxx, a r c s i n x ∼ x arcsinx\sim x arcsinxx, a r c t a n x ∼ x arctanx\sim x arctanxx, e x − 1 ∼ x e^x -1\sim x ex1x, I n ( 1 + x ) ∼ x In(1+x)\sim x In(1+x)x,
a x − 1 = e x I n a − 1 ∼ x I n a a^x-1= e^{xIna}-1\sim xIna ax1=exIna1xIna, 1 − c o s x ∼ 1 2 x 2 1-cosx\sim \frac{1}{2}x^2 1cosx21x2, ( 1 + x ) a − 1 ∼ a x (1+x)^a -1\sim ax (1+x)a1ax,
小 + 大 ∼ 大 小+大\sim 大 +, ∫ 0 x f ( t ) d t ∼ x \int_{0}^{x}f(t)dt\sim x 0xf(t)dtx

2 常用公式

2.1 和式夹逼准则的两个思路

n → ∞ : n ⋅ u m i n ≤ ∑ i = 1 n u i ≤ n ⋅ u m a x n → 有 限 : 1 ⋅ u m a x ≤ ∑ i = 1 n u i ≤ n ⋅ u m a x n\to \infty : n\cdot u_{min}\leq \sum\limits_{i=1}^{n}u_i \leq n\cdot u_{max}\\ n\to 有限 : 1\cdot u_{max}\leq \sum\limits_{i=1}^{n}u_i \leq n\cdot u_{max} n:numini=1nuinumaxn:1umaxi=1nuinumax

2.2 小基础

  1. ( a + b ) 3 = a 3 + 3 a 2 b + 3 a b 2 + b 3 (a+b)^3 = a^3+ 3a^2b + 3ab^2 + b^3 (a+b)3=a3+3a2b+3ab2+b3
  2. a 3 − b 3 = ( a − b ) ( a 2 + a b + b 3 ) a^3-b^3 = (a-b)(a^2 + ab + b^3) a3b3=(ab)(a2+ab+b3)
  3. ( a + b ) n = ∑ k = 0 n C n k a n − k b k (a+b)^n = \sum\limits_{k=0}^{n}C_n^ka^{n-k}b^k (a+b)n=k=0nCnkankbk
  4. ∑ k = 1 n k 2 = n ( n + 1 ) ( 2 n + 1 ) 6 \sum\limits_{k=1}^{n}k^2 = \frac{n(n+1)(2n+1)}{6} k=1nk2=6n(n+1)(2n+1)
  5. ∑ n = 1 ∞ 1 n 2 = π 2 6 \sum\limits_{n=1}^{\infty}\frac{1}{n^2} = \frac{\pi ^2}{6} n=1n21=6π2

2.3 一元二次方程解

X 1 , 2 = − b ± b 2 − 4 a c 2 a , X 1 + X 2 = − b a , X 1 X 2 = a c , 顶 点 : ( − b 2 a , c − b 2 4 a ) X_{1,2} = \frac{-b\pm \sqrt{b^2-4ac}}{2a}, X_1 + X_2 = -\frac{b}{a}, X_1X_2 = \frac{a}{c},\\顶点: (-\frac{b}{2a} , c-\frac{b^2}{4a}) X1,2=2ab±b24ac ,X1+X2=ab,X1X2=ca,:(2ab,c4ab2)

3 ★ \bigstar 常用展开公式 ★ \bigstar

  1. e x = 1 + x + x 2 2 ! + ⋯ = ∑ n = 0 ∞ x n n ! e^x = 1+x+\frac{x^2}{2!}+\cdots =\sum\limits_{n=0}^{\infty}\frac{x^n}{n!} ex=1+x+2!x2+=n=0n!xn
  2. I n ( 1 + x ) = x − x 2 2 + ⋯ = ∑ n = 0 ∞ ( − 1 ) n − 1 x n n ( − 1 < x ≤ 1 ) In(1+x) = x-\frac{x^2}{2}+\cdots =\sum\limits_{n=0}^{\infty}(-1)^{n-1}\frac{x^n}{n}\quad (-1In(1+x)=x2x2+=n=0(1)n1nxn(1<x1)
  3. I n ( 1 − x ) = − ∑ n = 0 ∞ x n n ( − 1 < x ≤ 1 ) In(1-x) = -\sum\limits_{n=0}^{\infty}\frac{x^n}{n}\quad (-1In(1x)=n=0nxn(1<x1)
  4. 1 1 − x = 1 + x + x 2 + ⋯ = ∑ n = 0 ∞ x n ∣ x ∣ < 1 \frac{1}{1-x} = 1+x+x^2+\cdots =\sum\limits_{n=0}^{\infty}x^n\quad \mid x\mid <1 1x1=1+x+x2+=n=0xnx<1
  5. s i n x = x − x 3 3 ! + ⋯ = ∑ n = 0 − ∞ ( − 1 ) n x 2 n + 1 ( 2 n + 1 ) ! sinx = x-\frac{x^3}{3!}+\cdots =\sum\limits_{n=0}^{-\infty}(-1)^{n}\frac{x^{2n+1}}{(2n+1)!} sinx=x3!x3+=n=0(1)n(2n+1)!x2n+1
  6. c o s x = 1 − x 2 2 ! + ⋯ = ∑ n = 0 − ∞ ( − 1 ) n x 2 n ( 2 n ) ! cosx = 1-\frac{x^2}{2!}+\cdots =\sum\limits_{n=0}^{-\infty}(-1)^{n}\frac{x^{2n}}{(2n)!} cosx=12!x2+=n=0(1)n(2n)!x2n
  7. t a n x = x + x 3 3 + O ( x 3 ) tanx = x+\frac{x^3}{3}+O(x^3) tanx=x+3x3+O(x3)
  8. a r c s i n x = x + x 3 6 + O ( x 3 ) arcsinx = x+\frac{x^3}{6}+O(x^3) arcsinx=x+6x3+O(x3)
  9. a r c t a n x = x − x 3 3 + O ( x 3 ) = ∑ n = 0 ∞ ( − 1 ) n x 2 n + 1 2 n + 1 arctanx = x-\frac{x^3}{3}+O(x^3) = \sum\limits_{n=0}^{\infty}(-1)^{n}\frac{x^{2n+1}}{2n+1} arctanx=x3x3+O(x3)=n=0(1)n2n+1x2n+1
  10. e x − e − x 2 = ∑ n = 0 ∞ x 2 n + 1 ( 2 n + 1 ) ! \frac{e^x-e^{-x}}{2} = \sum\limits_{n=0}^{\infty}\frac{x^{2n+1}}{(2n+1)!} 2exex=n=0(2n+1)!x2n+1
  11. e x + e − x 2 = ∑ n = 0 ∞ x 2 n ( 2 n ) ! \frac{e^x+e^{-x}}{2} = \sum\limits_{n=0}^{\infty}\frac{x^{2n}}{(2n)!} 2ex+ex=n=0(2n)!x2n
  12. ( 1 + x ) a = 1 + a x + a ( a − 1 ) 2 x 2 + O ( x 2 ) (1+x)^a = 1+ax+\frac{a(a-1)}2{x^2} + O(x^2) (1+x)a=1+ax+2a(a1)x2+O(x2)

4 常用不等式

  1. a r c t a n x < x < a r c s i n x ( 0 ≤ x ≤ 1 ) arctanx < x < arcsinx \quad (0\leq x\leq 1) arctanx<x<arcsinx(0x1)
  2. e x ≥ x + 1 ( ∀ x ) e^x \geq x+1\quad (∀x) exx+1(x)
  3. x − 1 ≥ I n x ( x > 0 ) x-1 \geq Inx\quad (x>0) x1Inx(x>0)
  4. x > s i n x ( x > 0 ) x> sinx\quad (x>0) x>sinx(x>0)
  5. 1 1 + x < I n ( 1 + 1 x ) < 1 x \frac{1}{1+x}1+x1<In(1+x1)<x1
  6. x 1 + x < I n ( 1 + x ) < x \frac{x}{1+x}1+xx<In(1+x)<x
  7. a b ≤ a + b 2 ≤ a 2 + b 2 2 ( a , b > 0 ) \sqrt{ab}\leq \frac{a+b}{2}\leq \sqrt{\frac{a^2+b^2}{2}}\quad (a,b>0) ab 2a+b2a2+b2 (a,b>0)
  8. a b c 3 ≤ a + b + c 3 ( a , b , c > 0 ) \sqrt[3]{abc}\leq \frac{a+b+c}{3}\quad (a,b,c>0) 3abc 3a+b+c(a,b,c>0)
  9. ∣ a ± b ∣ ≤ ∣ a ∣ + ∣ b ∣ \mid a\pm b\mid \leq \mid a\mid + \mid b\mid a±ba+b
  10. ∣ ∣ a ∣ − ∣ b ∣ ∣ ≤ ∣ a − b ∣ \mid \mid a\mid - \mid b\mid \mid \leq \mid a-b\mid abab
  11. ∣ ∫ a b f ( x ) d x ∣ ≤ ∫ a b ∣ f ( x ) ∣ d x \mid \int_a^bf(x)dx\mid \leq \int_a^b\mid f(x)\mid dx abf(x)dxabf(x)dx

5 三角变换

  1. 诱导公式法则:奇变偶不变,符号看象限!
  2. s i n 2 x = 2 s i n x c o s x sin2x = 2sinxcosx sin2x=2sinxcosx
  3. c o s 2 x = c o s 2 x − s i n 2 x = 1 − 2 s i n 2 x = 2 c o s 2 − 1 cos2x = cos^2x - sin^2x = 1-2sin^2x = 2cos^2-1 cos2x=cos2xsin2x=12sin2x=2cos21
  4. s i n 3 x = − 4 s i n 3 x + 3 s i n x sin3x = -4sin^3x + 3sinx sin3x=4sin3x+3sinx
  5. c o s 3 x = 4 c o s 2 x − 3 c o s x cos3x = 4cos^2x - 3cosx cos3x=4cos2x3cosx
  6. s i n x ⋅ c o s y = 1 2 [ s i n ( x + y ) + s i n ( x − y ) ] sinx\cdot cosy = \frac{1}{2}[sin(x+y)+sin(x-y)] sinxcosy=21[sin(x+y)+sin(xy)]
  7. s i n 2 x 2 = 1 2 ( 1 − c o s x ) sin^2\frac{x}{2} = \frac{1}{2}(1-cosx) sin22x=21(1cosx)
  8. c o s 2 x 2 = 1 2 ( 1 + c o s x ) cos^2\frac{x}{2} = \frac{1}{2}(1+cosx) cos22x=21(1+cosx)
  9. t a n 2 x 2 = 1 − c o s x s i n x = s i n x 1 + c o s x tan^2\frac{x}{2} = \frac{1-cosx}{sinx} = \frac{sinx}{1+cosx} tan22x=sinx1cosx=1+cosxsinx
  10. s i n x = 2 t a n x 2 1 + t a n 2 x 2 sinx = \frac{2tan\frac{x}{2}}{1+tan^2\frac{x}{2}} sinx=1+tan22x2tan2x
  11. c o s x = 1 − t a n 2 x 2 1 + t a n 2 x 2 cosx = \frac{1-tan^2\frac{x}{2}}{1+tan^2\frac{x}{2}} cosx=1+tan22x1tan22x
  12. t a n 2 x = 2 t a n x 1 − t a n 2 x tan2x = \frac{2tanx}{1-tan^2x} tan2x=1tan2x2tanx
  13. c o t 2 x = c o t 2 x − 1 2 c o t x cot2x = \frac{cot^2x - 1}{2cotx} cot2x=2cotxcot2x1
  14. 1 + t a n 2 x = s e c 2 x 1+tan^2x = sec^2x 1+tan2x=sec2x
  15. 1 + c o t 2 x = c s c 2 x 1+cot^2x = csc^2x 1+cot2x=csc2x

6 ★ \bigstar 微分 ★ \bigstar

6.1 定义式

  • f ′ ( x 0 ) = lim ⁡ Δ x → 0 f ( x 0 + Δ x ) − f ( x 0 ) Δ x = lim ⁡ x → x 0 f ( x ) − f ( x 0 ) x − x 0 f'(x_0) = \lim\limits_{\Delta x\to{0}}\frac{f(x_0 + \Delta x) - f(x_0)}{\Delta x} = \lim\limits_{x\to{x_0}}\frac{f(x) - f(x_0)}{x-x_0} f(x0)=Δx0limΔxf(x0+Δx)f(x0)=xx0limxx0f(x)f(x0)
  • f ( n ) ( x 0 ) = lim ⁡ x → x 0 f ( n − 1 ) ( x ) − f ( n − 1 ) ( x 0 ) x − x 0 f^{(n)}(x_0) = \lim\limits_{x\to{x_0}} \frac{f^{(n-1)}(x) - f^{(n-1)}(x_0)}{x-x_0} f(n)(x0)=xx0limxx0f(n1)(x)f(n1)(x0)
  • 连续 ⇏ \nRightarrow 可导,可导 ⇒ \Rightarrow 连续,可导 ⇔ \Leftrightarrow 可微

6.2 常用难记微分公式

  1. ( t a n x ) ′ = s e c 2 x (tanx)' = sec^2x (tanx)=sec2x
  2. ( c o t x ) ′ = − c s c 2 x (cotx)' = -csc^2x (cotx)=csc2x
  3. ( s e c x ) ′ = s e c x t a n x , ( c s c x ) ′ = − c s c x c o t x (secx)' = secxtanx, (cscx)' = -cscxcotx (secx)=secxtanx,(cscx)=cscxcotx
  4. ( a r c s i n x ) ′ = 1 1 − x 2 , ( a r c c o s x ) ′ = − 1 1 − x 2 (arcsinx)' = \frac{1}{\sqrt{1-x^2}},\quad (arccosx)' = -\frac{1}{\sqrt{1-x^2}} (arcsinx)=1x2 1,(arccosx)=1x2 1
  5. ( a r c t a n x ) ′ = 1 1 + x 2 , ( a r c c o t x ) ′ = − 1 1 + x 2 (arctanx)' = \frac{1}{1+x^2},\quad (arccotx)' = -\frac{1}{1+x^2} (arctanx)=1+x21,(arccotx)=1+x21
  6. ( I n ∣ c o s x ∣ ) ′ = − t a n x (In\mid cosx\mid)' = -tanx (Incosx)=tanx
  7. ( I n ∣ s i n x ∣ ) ′ = c o t x (In\mid sinx\mid)' = cotx (Insinx)=cotx
  8. ( I n ∣ s e c x + t a n x ∣ ) ′ = s e c x (In\mid secx + tanx\mid)' = secx (Insecx+tanx)=secx
  9. ( I n ∣ c s c x − c o t x ∣ ) ′ = c s c x (In\mid cscx - cotx\mid)' = cscx (Incscxcotx)=cscx
  10. [ I n ( x + x 2 ± a 2 ) ] ′ = 1 x 2 ± a 2 [In(x+\sqrt{x^2\pm a^2})]' = \frac{1}{\sqrt{x^2\pm a^2}} [In(x+x2±a2 )]=x2±a2 1
  11. d x 2 = ( d x ) 2 dx^2 = (dx)^2 dx2=(dx)2
  12. d ( x 2 ) = 2 x d x d(x^2) = 2xdx d(x2)=2xdx
  13. ( u v w ) ′ = u ′ v w + u v ′ w + u v w ′ (uvw)' = u'vw+uv'w+uvw' (uvw)=uvw+uvw+uvw
  14. ( u v ) ( n ) = ∑ k = 0 n C n k u ( n − k ) v ( k ) (uv)^{(n)} = \sum\limits_{k=0}^{n}C_n^ku^{(n-k)}v^{(k)} (uv)(n)=k=0nCnku(nk)v(k)

6.3 分析函数关注点

定义域、奇偶性、对称性、图形变换、单调性、极值、最值、凹凸性、拐点、三种渐近线(铅垂、水平、斜)

7 ★ \bigstar 积分 ★ \bigstar

7.1 定积分定义

  • ∫ a b f ( x ) d x = lim ⁡ n → ∞ ∑ i = 1 ∞ f ( a + b − a n i ) b − a n \int_a^bf(x)dx =\lim\limits_{n\to{\infty}} \sum\limits_{i=1}^{\infty} f(a+\frac{b-a}{n}i)\frac{b-a}{n} abf(x)dx=nlimi=1f(a+nbai)nba
  • ∫ 0 1 f ( x ) d x = lim ⁡ n → ∞ ∑ i = 1 ∞ f ( i n ) 1 n \int_0^1f(x)dx =\lim\limits_{n\to{\infty}} \sum\limits_{i=1}^{\infty} f(\frac{i}{n})\frac{1}{n} 01f(x)dx=nlimi=1f(ni)n1
  • ∫ 0 x f ( x ) d x = lim ⁡ n → ∞ ∑ i = 1 ∞ f ( x n i ) x n \int_0^xf(x)dx =\lim\limits_{n\to{\infty}} \sum\limits_{i=1}^{\infty} f(\frac{x}{n}i)\frac{x}{n} 0xf(x)dx=nlimi=1f(nxi)nx

7.2 基本积分表

  1. ∫ x k d x = 1 k + 1 x k + 1 + C \int x^k dx = \frac{1}{k+1}x^{k+1} + C xkdx=k+11xk+1+C
  2. ∫ a x d x = a x I n a + C \int a^x dx = \frac{a^x}{Ina} +C axdx=Inaax+C
  3. ∫ s i n x d x = − c o s + C \int sinx dx = -cos +C sinxdx=cos+C
  4. ∫ c o s d x = s i n x + C \int cos dx = sinx +C cosdx=sinx+C
  5. ∫ t a n x d x = − I n ∣ c o s x ∣ + C \int tanx dx = -In\mid cosx\mid +C tanxdx=Incosx+C
  6. ∫ c o t x d x = I n ∣ s i n x ∣ + C \int cotx dx = In\mid sinx\mid +C cotxdx=Insinx+C
  7. ∫ s e c x d x = I n ∣ s e c x + t a n x ∣ + C \int secx dx = In\mid secx + tanx\mid +C secxdx=Insecx+tanx+C
  8. ∫ c s c x d x = I n ∣ c s c x − c o t x ∣ + C \int cscx dx = In\mid cscx - cotx\mid +C cscxdx=Incscxcotx+C
  9. ∫ s e c 2 x d x = t a n x + C \int sec^2x dx = tanx +C sec2xdx=tanx+C
  10. ∫ c s c 2 x d x = − c o t x + C \int csc^2x dx = -cotx +C csc2xdx=cotx+C
  11. ∫ s e c x t a n x d x = s e c x + C \int secxtanx dx = secx +C secxtanxdx=secx+C
  12. ∫ c s c x c o t x d x = − c s c x + C \int cscxcotx dx = -cscx +C cscxcotxdx=cscx+C
  13. ∫ 1 1 − x 2 d x = a r c s i n x + C \int \frac{1}{\sqrt{1-x^2}} dx = arcsinx +C 1x2 1dx=arcsinx+C
  14. ∫ 1 a 2 − x 2 d x = a r c s i n x a + C \int \frac{1}{\sqrt{a^2-x^2}} dx = arcsin\frac{x}{a} +C a2x2 1dx=arcsinax+C
  15. ∫ 1 1 + x 2 d x = a r c t a n x + C \int \frac{1}{1+x^2} dx = arctanx +C 1+x21dx=arctanx+C
  16. ∫ 1 a 2 + x 2 d x = 1 a a r c t a n x a + C ( a > 0 ) \int \frac{1}{a^2+x^2} dx = \frac{1}{a}arctan\frac{x}{a} +C\quad (a>0) a2+x21dx=a1arctanax+C(a>0)
  17. ∫ 1 x 2 + a 2 d x = I n ( x + x 2 + a 2 ) + C \int \frac{1}{\sqrt{x^2+a^2}} dx = In(x+\sqrt{x^2+a^2}) +C x2+a2 1dx=In(x+x2+a2 )+C
  18. ∫ 1 x 2 − a 2 d x = I n ( x + x 2 − a 2 ) + C ( ∣ x ∣ > ∣ a ∣ ) \int \frac{1}{\sqrt{x^2-a^2}} dx = In(x+\sqrt{x^2-a^2}) +C\quad (\mid x\mid>\mid a\mid) x2a2 1dx=In(x+x2a2 )+C(x>a)
  19. ∫ 1 x 2 − a 2 d x = 1 2 a I n ∣ x − a x + a ∣ + C \int \frac{1}{x^2-a^2} dx = \frac{1}{2a}In\mid \frac{x-a}{x+a}\mid +C x2a21dx=2a1Inx+axa+C
  20. ∫ 1 a 2 − x 2 d x = 1 2 a I n ∣ x + a x − a ∣ + C \int \frac{1}{a^2-x^2} dx = \frac{1}{2a}In\mid \frac{x+a}{x-a}\mid +C a2x21dx=2a1Inxax+a+C
  21. ∫ a 2 − x 2 d x = a 2 2 a r c s i n x a + x 2 a 2 − x 2 + C ( ∣ x ∣ < a ) \int \sqrt{a^2-x^2} dx = \frac{a^2}{2}arcsin\frac{x}{a} + \frac{x}{2}\sqrt{a^2-x^2} +C\quad (\mid x\mida2x2 dx=2a2arcsinax+2xa2x2 +C(x<a)
  22. ∫ s i n 2 x d x = x 2 − s i n 2 x 4 + C \int sin^2x dx = \frac{x}{2} -\frac{sin2x}{4} +C sin2xdx=2x4sin2x+C
  23. ∫ c o s 2 x d x = x 2 + s i n 2 x 4 + C \int cos^2x dx = \frac{x}{2} +\frac{sin2x}{4} +C cos2xdx=2x+4sin2x+C

7.3 常用积分公式

  1. ∫ a b f ( x ) d x = ∫ a b f ( a + b − x ) d x \int_a^b f(x) dx = \int_a^b f(a+b-x) dx abf(x)dx=abf(a+bx)dx
  2. ∫ a b f ( x ) d x = 1 2 ∫ a b [ f ( x ) + f ( a + b − x ) ] d x \int_a^b f(x) dx = \frac{1}{2} \int_a^b [f(x)+f(a+b-x)] dx abf(x)dx=21ab[f(x)+f(a+bx)]dx
  3. ∫ a b f ( x ) d x = ∫ a a + b 2 [ f ( x ) + f ( a + b − x ) ] d x \int_a^b f(x) dx = \int_a^{\frac{a+b}{2}} [f(x)+f(a+b-x)] dx abf(x)dx=a2a+b[f(x)+f(a+bx)]dx
  4. 点火公式 ∫ 0 π 2 s i n n x d x = ∫ 0 π 2 c o s n x d x = n − 1 n n − 3 n − 2 ⋯ \int_0^\frac{\pi}{2}sin^nxdx = \int_0^\frac{\pi}{2}cos^nxdx = \frac{n-1}{n} \frac{n-3}{n-2} \cdots 02πsinnxdx=02πcosnxdx=nn1n2n3
  5. ∫ 0 π x f ( s i n x ) d x = π 2 ∫ 0 π f ( s i n x ) d x = π ∫ 0 π 2 f ( s i n x ) \int_0^{\pi} xf(sinx) dx = \frac{\pi}{2}\int_0^{\pi} f(sinx) dx =\pi \int_0^{\frac{\pi}{2}} f(sinx) 0πxf(sinx)dx=2π0πf(sinx)dx=π02πf(sinx)
  6. ∫ 0 π 2 f ( s i n x ) d x = ∫ 0 π 2 f ( c o s x ) d x \int_0^{\frac{\pi}{2}} f(sinx) dx = \int_0^{\frac{\pi}{2}} f(cosx) dx 02πf(sinx)dx=02πf(cosx)dx
  7. ∫ 0 π 2 f ( s i n x , c o s x ) d x = ∫ 0 π 2 f ( c o s x , s i n x ) d x \int_0^{\frac{\pi}{2}} f(sinx, cosx) dx = \int_0^{\frac{\pi}{2}} f(cosx,sinx) dx 02πf(sinx,cosx)dx=02πf(cosx,sinx)dx
  8. ∫ a b f ( x ) d x = ∫ − π 2 π 2 f ( a + b 2 + b − a 2 s i n t ) ⋅ b − a 2 c o s t d t , ( x − a + b 2 = b − a 2 s i n t ) \int_a^b f(x) dx = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} f(\frac{a+b}{2} + \frac{b-a}{2} sint)\cdot \frac{b-a}{2}cost dt, \quad (x-\frac{a+b}{2} = \frac{b-a}{2}sint) abf(x)dx=2π2πf(2a+b+2basint)2bacostdt,(x2a+b=2basint)
  9. ∫ a b f ( x ) d x = ∫ 0 1 ( b − a ) f [ a + ( b − a ) t ] d t , ( x − a = ( b − a ) t ) \int_a^b f(x) dx = \int_{0}^{1} (b-a) f[a+(b-a)t] dt, \quad (x-a = (b-a)t) abf(x)dx=01(ba)f[a+(ba)t]dt,(xa=(ba)t)
  10. ∫ − a a f ( x ) d x = ∫ 0 a [ f ( x ) + f ( − x ) ] d x \int_{-a}^a f(x) dx = \int_{0}^{a} [f(x) + f(-x)] dx aaf(x)dx=0a[f(x)+f(x)]dx
  11. ∫ 0 n π x ∣ s i n x ∣ d x = n 2 π \int_0^{n\pi} x\mid sinx\mid dx = n^2\pi 0nπxsinxdx=n2π

7.4 积分常用方法

凑微分、换元、分部积分、通分

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