由于公式太多，汇总成一篇容易打不开，所以分三篇。整篇链接：https://blog.csdn.net/zhaohongfei_358/article/details/106039576

a n + 1 = f ( a n ) 结 论 一 ： f ′ ( x ) > 0 ， { a 2 > a 1    ⟹    { a n } ↗ 单 调 递 增 a 2 < a 1    ⟹    { a n } ↘ 单 调 递 减 结 论 二 （ 压 缩 映 像 原 理 ） ： ∃ k ∈ ( 0 , 1 ) ， 使 得 ∣ f ′ ( x ) ∣ ≤ k    ⟹    a n 收 敛 \begin{aligned} & a_{n+1} = f(a_n) \\\\ 结论一： & f'(x) > 0 ， \begin{cases} a_2 > a_1 \implies \{ a_n \} \nearrow单调递增 \\ a_2 < a_1 \implies \{ a_n \} \searrow单调递减 \end{cases} \\\\ 结论二（压缩映像原理）：& \exist k \in (0,1)，使得 |f'(x)| \le k \implies {a_n} 收敛 \end{aligned} 结论一：结论二（压缩映像原理）：​an+1​=f(an​)f′(x)>0，{a2​>a1​⟹{an​}↗单调递增a2​f[g(x)]}′=f′[g(x)]g′(x)

y = f ( x ) , x = φ ( y )    ⟹    φ ′ ( y ) = 1 f ′ ( x ) y x ′ = d y d x = 1 d x d y = 1 x y ′ y x x ′ ′ = d 2 y d x 2 = d ( d y d x ) d x = d ( 1 x y ′ ) d x = d ( 1 x y ′ ) d y ⋅ d y d x = d ( 1 x y ′ ) d y ⋅ 1 x y ′ = − x y y ′ ′ ( x y ′ ) 3 \begin{aligned} & y = f(x), x = \varphi(y) \implies \varphi ' (y) = \frac{1}{f'(x)} \\ \\ & y'_x = \frac{dy}{dx} = \frac{1}{\frac{dx}{dy}} = \frac{1}{x'_y} \\ \\ & y^{''}_{xx} = \frac{d^2 y}{dx^2} = \frac{d(\frac{dy}{dx})}{dx} = \frac{d(\frac{1}{x'_y})}{dx} = \frac{d(\frac{1}{x'_y})}{dy} \cdot \frac{dy}{dx} = \frac{d(\frac{1}{x'_y})}{dy} \cdot \frac{1}{x'_y} = \frac{-x^{''}_{yy}}{(x'_y)^3} \end{aligned} ​y=f(x),x=φ(y)⟹φ′(y)=f′(x)1​yx′​=dxdy​=dydx​1​=xy′​1​yxx′′​=dx2d2y​=dxd(dxdy​)​=dxd(xy′​1​)​=dyd(xy′​1​)​⋅dxdy​=dyd(xy′​1​)​⋅xy′​1​=(xy′​)3−xyy′′​​​

{ x = φ ( t ) y = ψ ( t ) d y d x = d y / d t d x / d t = ψ ′ ( t ) φ ′ ( t ) d 2 y d x 2 = d ( d y d x ) d x = d ( d y d x ) / d t d x / d t = ψ ′ ′ ( t ) φ ′ ( t ) − ψ ′ ( t ) φ ′ ′ ( t ) [ φ ′ ( t ) ] 3 \begin{aligned} & \begin{cases} x = \varphi (t) \\ y = \psi (t) \end{cases} \\\\ & \frac{dy}{dx} = \frac{dy/dt}{dx/dt} = \frac{\psi ' (t)}{\varphi ' (t)} \\ \\ & \frac{d^2 y}{dx^2} = \frac{d(\frac{dy}{dx})}{dx} = \frac {d(\frac{dy}{dx})/dt}{dx/dt} = \frac{\psi '' (t) \varphi '(t) - \psi '(t) \varphi '' (t) }{[\varphi ' (t)]^3} \end{aligned} ​{x=φ(t)y=ψ(t)​dxdy​=dx/dtdy/dt​=φ′(t)ψ′(t)​dx2d2y​=dxd(dxdy​)​=dx/dtd(dxdy​)/dt​=[φ′(t)]3ψ′′(t)φ′(t)−ψ′(t)φ′′(t)​​

设 F ( x ) = ∫ φ 1 ( x ) φ 2 ( x ) f ( t ) d t , 则 F ′ ( x ) = d d x [ ∫ φ 1 ( x ) φ 2 ( x ) f ( t ) d t ] = f [ φ 2 ( x ) ] φ 2 ′ ( x ) − f [ φ 1 ( x ) ] φ 1 ′ ( x ) \begin{aligned} & 设 F(x) = \int ^{\varphi_2(x)}_{\varphi_1(x)} f(t) dt, 则 \\ \\ & F'(x) = \frac{d}{dx}\begin{bmatrix}\int ^{\varphi_2(x)}_{\varphi_1(x)} f(t) dt \end{bmatrix} = f[\varphi _2(x)]\varphi '_2(x) - f[\varphi_1(x)]\varphi '_1(x) \end{aligned} ​设F(x)=∫φ1​(x)φ2​(x)​f(t)dt,则F′(x)=dxd​[∫φ1​(x)φ2​(x)​f(t)dt​]=f[φ2​(x)]φ2′​(x)−f[φ1​(x)]φ1′​(x)​

( x a ) ′ = a x a − 1       ( a 为 常 数 ) ( a x ) ′ = a x ln ⁡ a ( e x ) ′ = e x ( l o g a x ) ′ = 1 x ln ⁡ a       ( a > 0 , a ≠ 1 ) ( ln ⁡ x ) ′ = 1 x ( sin ⁡ x ) ′ = cos ⁡ x ( cos ⁡ x ) ′ = − sin ⁡ x ( arcsin ⁡ x ) ′ = 1 1 − x 2 ( arccos ⁡ x ) ′ = − 1 1 − x 2 ( tan ⁡ x ) ′ = sec ⁡ 2 x ( cot ⁡ x ) ′ = − csc ⁡ 2 x ( arctan ⁡ x ) ′ = 1 1 + x 2 ( a r c c o t   x ) ′ = − 1 1 + x 2 ( sec ⁡ x ) ′ = sec ⁡ x ⋅ tan ⁡ x ( csc ⁡ x ) ′ = − csc ⁡ x ⋅ cot ⁡ x [ ln ⁡ ( x + x 2 + 1 ) ] ′ = 1 x 2 + 1 [ ln ⁡ ( x + x 2 − 1 ) ] ′ = 1 x 2 − 1 \begin{aligned} & (x^a)' = a x^{a-1} ~~~~~(a为常数) \\ \\ & (a^x)' = a^x \ln a \\ \\ & (e^x)' = e^x \\ \\ & (log_a x)' = \frac{1}{x \ln a} ~~~~~(a>0, a \ne 1) \\ \\ & (\ln x)' = \frac{1}{x} \\ \\ & (\sin x)' = \cos x \\ \\ & (\cos x)' = -\sin x \\ \\ & (\arcsin x)' = \frac{1}{\sqrt{1-x^2}} \\ \\ & (\arccos x)' = - \frac{1}{\sqrt{1-x^2}} \\ \\ & (\tan x)' = \sec ^2 x \\ \\ & (\cot x)' = - \csc^2 x \\ \\ & (\arctan x)' = \frac{1}{1+x^2} \\ \\ & (arccot ~ x)' = - \frac{1}{1+x^2} \\ \\ & (\sec x)' = \sec x \cdot \tan x \\ \\ & (\csc x)' = - \csc x \cdot \cot x \\ \\ & [\ln (x+\sqrt{x^2+1})]' = \frac{1}{\sqrt{x^2+1}} \\ \\ & [\ln (x+\sqrt{x^2-1})]' = \frac{1}{\sqrt{x^2-1}} \\ \\ \end{aligned} ​(xa)′=axa−1     (a为常数)(ax)′=axlna(ex)′=ex(loga​x)′=xlna1​     (a>0,a​=1)(lnx)′=x1​(sinx)′=cosx(cosx)′=−sinx(arcsinx)′=1−x2 ​1​(arccosx)′=−1−x2 ​1​(tanx)′=sec2x(cotx)′=−csc2x(arctanx)′=1+x21​(arccot x)′=−1+x21​(secx)′=secx⋅tanx(cscx)′=−cscx⋅cotx[ln(x+x2+1 ​)]′=x2+1 ​1​[ln(x+x2−1 ​)]′=x2−1 ​1​​

[ u ± v ] ( n ) = u ( n ) ± v ( n ) [u \pm v ]^{(n)} = u^{(n)} \pm v^{(n)} [u±v](n)=u(n)±v(n) ( u v ) ( n ) = u ( n ) v + C n 1 u ( n − 1 ) v ′ + C n 2 u ( n − 2 ) v ′ ′ + . . . + C n k u ( n − k ) v ( k ) + . . . + C n n − 1 u ′ v ( n − 1 ) + u v ( n ) = ∑ k = 0 n C n k u ( n − k ) v ( k ) \begin{aligned} (uv)^{(n)} & = u^{(n)}v + C_n^1 u^{(n-1)}v' + C_n^2 u^{(n-2)}v'' + ... + C_n^k u^{(n-k)}v^{(k)} + ... + C_n^{n-1} u'v^{(n-1)} + uv^{(n)} \\ & = \displaystyle\sum_{k=0}^n C_n^k u^{(n-k)}v^{(k)} \end{aligned} (uv)(n)​=u(n)v+Cn1​u(n−1)v′+Cn2​u(n−2)v′′+...+Cnk​u(n−k)v(k)+...+Cnn−1​u′v(n−1)+uv(n)=k=0∑n​Cnk​u(n−k)v(k)​

( a x ) ( n ) = a x ( ln ⁡ a ) n ( e x ) ( n ) = e x ( sin ⁡ k x ) ( n ) = k n sin ⁡ ( k x + n ⋅ π 2 ) ( cos ⁡ k x ) ( n ) = k n cos ⁡ ( k x + n ⋅ π 2 ) ( ln ⁡ x ) ( n ) = ( − 1 ) n − 1 ( n − 1 ) ! x n       ( x > 0 ) [ ln ⁡ ( 1 + x ) ] ( n ) = ( − 1 ) n − 1 ( n − 1 ) ! ( x + 1 ) n       ( x > − 1 ) [ ( x + x 0 ) m ] ( n ) = m ( m − 1 ) ( m − 2 ) ⋅ ⋅ ⋅ ⋅ ( m − n + 1 ) ( x + x 0 ) m − n ( 1 x + a ) ( n ) = ( − 1 ) n ⋅ n ! ( x + a ) n + 1 \begin{aligned} & (a^x)^{(n)} = a^x (\ln a)^n \\ \\ & (e^x)^{(n)} = e^x \\ \\ & (\sin kx)^{(n)} = k^n \sin (kx + n\cdot \frac{\pi}{2}) \\ \\ & (\cos kx)^{(n)} = k^n \cos (kx + n\cdot \frac{\pi}{2}) \\ \\ & (\ln x) ^ {(n)} = (-1)^{n-1} \frac{(n-1)!}{x^n} ~~~~~(x>0) \\ \\ & [\ln(1+x)]^{(n)} = (-1)^{n-1} \frac{(n-1)!}{(x+1)^n} ~~~~~(x>-1) \\ \\ & [(x+x_0)^m]^{(n)} = m(m-1)(m-2)\cdotp\cdotp\cdotp\cdot (m-n+1)(x+x_0)^{m-n} \\ \\ & (\frac{1}{x+a})^{(n)} = \frac{(-1)^n \cdot n!}{(x+a)^{n+1}} \\ \\ \end{aligned} ​(ax)(n)=ax(lna)n(ex)(n)=ex(sinkx)(n)=knsin(kx+n⋅2π​)(coskx)(n)=kncos(kx+n⋅2π​)(lnx)(n)=(−1)n−1xn(n−1)!​     (x>0)[ln(1+x)](n)=(−1)n−1(x+1)n(n−1)!​     (x>−1)[(x+x0​)m](n)=m(m−1)(m−2)⋅⋅⋅⋅(m−n+1)(x+x0​)m−n(x+a1​)(n)=(x+a)n+1(−1)n⋅n!​​

{ 1.    f ′ ( x ) 左 右 异 号    ⟹    { 左 正 右 负    ⟹    极 大 值 左 负 右 正    ⟹    极 小 值 2.    f ′ ( x ) = 0 , f ′ ′ ( x ) ≠ 0    ⟹    { f ′ ′ ( x ) < 0    ⟹    极 大 值 f ′ ′ ( x ) > 0    ⟹    极 小 值 3.    f ′ ′ ( x ) 到 f ( n − 1 ) ( x ) = 0 ， f ( n ) ( x ) ≠ 0 ， n 为 偶 数    ⟹    { f ( n ) ( x ) < 0    ⟹    极 大 值 f ( n ) ( x ) > 0    ⟹    极 小 值 \begin{cases} 1.~~f'(x)左右异号 \implies \begin{cases} 左正右负 \implies 极大值 \\ 左负右正 \implies 极小值 \end{cases} \\ \\ 2.~~f'(x)=0, f''(x)\ne 0 \implies \begin{cases} f''(x) < 0 \implies 极大值 \\ f''(x)>0 \implies 极小值 \end{cases} \\ \\ 3. ~~f''(x) 到 f^{(n-1)}(x)=0 ，f^{(n)}(x) \ne 0， n为偶数 \implies \begin{cases} f^{(n)}(x) < 0 \implies 极大值 \\ f^{(n)}(x) > 0 \implies 极小值 \end{cases} \end{cases} ⎩⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎧​1.  f′(x)左右异号⟹{左正右负⟹极大值左负右正⟹极小值​2.  f′(x)=0,f′′(x)​=0⟹{f′′(x)0⟹极小值​3.  f′′(x)到f(n−1)(x)=0，f(n)(x)​=0，n为偶数⟹{f(n)(x)0⟹极小值​​

1. { f ( x 1 + x 2 2 ) < f ( x 1 ) + f ( x 2 ) 2    ⟹    凹 f ( x 1 + x 2 2 ) > f ( x 1 ) + f ( x 2 ) 2    ⟹    凸 2. { f ′ ′ ( x ) > 0    ⟹    凹 f ′ ′ ( x ) < 0    ⟹    凸 \begin{aligned} 1.&\begin{cases} f(\frac{x_1+x_2}{2}) < \frac{f(x_1)+f(x_2)}{2} \implies 凹 \\\\ f(\frac{x_1+x_2}{2}) > \frac{f(x_1)+f(x_2)}{2} \implies 凸 \end{cases} \\\\ 2.&\begin{cases} f''(x) > 0 \implies 凹 \\\\ f''(x) < 0 \implies 凸 \end{cases} \end{aligned} 1.2.​⎩⎪⎨⎪⎧​f(2x1​+x2​​)2f(x1​)+f(x2​)​⟹凸​⎩⎪⎨⎪⎧​f′′(x)>0⟹凹f′′(x) 0    ⟹    凸 → 凹 3.    f ′ ′ ( x ) 到 f ( n − 1 ) ( x ) = 0 ， f ( n ) ( x ) ≠ 0 ， n 为 奇 数    ⟹    { f ( n ) ( x ) < 0    ⟹    凹 → 凸 f ( n ) ( x ) > 0    ⟹    凸 → 凹 \begin{cases} 1.~~f''(x)左右异号 \implies \begin{cases} 左负右正 \implies 凸 \to 凹 \\ 左正右负 \implies 凹 \to 凸 \end{cases} \\ \\ 2.~~f''(x)=0, f'''(x)\ne 0 \implies \begin{cases} f'''(x) < 0 \implies 凹 \to 凸 \\ f'''(x)>0 \implies 凸 \to 凹 \end{cases} \\ \\ 3. ~~f''(x) 到 f^{(n-1)}(x)=0 ，f^{(n)}(x) \ne 0， n为奇数 \implies \begin{cases} f^{(n)}(x) < 0 \implies 凹 \to 凸 \\ f^{(n)}(x) > 0 \implies 凸 \to 凹 \end{cases} \end{cases} ⎩⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎧​1.  f′′(x)左右异号⟹{左负右正⟹凸→凹左正右负⟹凹→凸​2.  f′′(x)=0,f′′′(x)​=0⟹{f′′′(x)0⟹凸→凹​3.  f′′(x)到f(n−1)(x)=0，f(n)(x)​=0，n为奇数⟹{f(n)(x)0⟹凸→凹​​

lim ⁡ x → + ∞ f ( x ) x = a      lim ⁡ x → + ∞ ( f ( x ) − a x ) = b    ⟹    斜 渐 近 线 为 ： y = a x + b \lim_{x \to +\infty} \frac{f(x)}{x} = a ~~~~\lim_{x \to + \infty}(f(x)-ax) = b \implies 斜渐近线为： y=ax+b x→+∞lim​xf(x)​=a    x→+∞lim​(f(x)−ax)=b⟹斜渐近线为：y=ax+b

密 切 圆 半 径       r = ( 1 + y ′ 2 ) 3 2 ∣ y ′ ′ ∣ 曲 率       K = 1 r = ∣ y ′ ′ ∣ ( 1 + y ′ 2 ) 3 2 曲 率 圆       ( X − ( x − y ′ ( 1 + y ′ 2 ) y ′ ′ 2 ) ) 2 + ( Y − ( y + 1 + y ′ 2 y ′ ′ 2 ) ) = ( ( 1 + y ′ 2 ) 3 2 ∣ y ′ ′ ∣ ) 2 \begin{aligned} 密切圆半径 ~~~~~ & r = \frac{(1+y'^2)^{\frac{3}{2}}}{|y''|} \\ \\ 曲率 ~~~~~ &K = \frac{1}{r} = \frac{|y''|}{(1+y'^2)^{\frac{3}{2}}} \\ \\ 曲率圆 ~~~~~& (X-(x-\frac{y'(1+y'^2)}{y''^2}))^2 +(Y-(y+\frac{1+y'^2}{y''^2})) = (\frac{(1+y'^2)^{\frac{3}{2}}}{|y''|})^2 \end{aligned} 密切圆半径     曲率     曲率圆     ​r=∣y′′∣(1+y′2)23​​K=r1​=(1+y′2)23​∣y′′∣​(X−(x−y′′2y′(1+y′2)​))2+(Y−(y+y′′21+y′2​))=(∣y′′∣(1+y′2)23​​)2​

∫ a b f ( x ) d x = lim ⁡ n → ∞ ∑ i = 1 n f ( a + b − a n i ) b − a n 常 用 ： ∫ 0 1 f ( x ) d x = lim ⁡ n → ∞ ∑ i = 1 n f ( i n ) ⋅ 1 n ∫ 0 k f ( x ) d x = lim ⁡ n → ∞ ∑ i = 1 k n f ( i n ) ⋅ 1 n 二 重 定 积 分 精 确 定 义 ： ∬ D f ( x , y ) d σ = lim ⁡ n → ∞ ∑ i = 1 n ∑ j = 1 n f ( a + b − a n i , c + d − c n j ) ⋅ b − a n ⋅ d − c n 常 用 ： ∫ 0 1 ∫ 0 1 f ( x , y ) d x d y = lim ⁡ n → ∞ ∑ i = 1 n ∑ j = 1 n f ( i n , j n ) ⋅ 1 n 2 \begin{aligned} & \int_a^b f(x) dx = \lim_{n \to \infty} \displaystyle\sum_{i=1}^n f(a+\frac{b-a}{n}i)\frac{b-a}{n} \\ \\ \\ 常用：& \int_0^1 f(x) dx = \lim_{n \to \infty} \displaystyle\sum_{i=1}^n f(\frac{i}{n})\cdot\frac{1}{n} \\ \\ & \int_0^k f(x) dx = \lim_{n \to \infty} \displaystyle\sum_{i=1}^{kn} f(\frac{i}{n})\cdot\frac{1}{n} \\ \\ \\ 二重定积分精确定义：& \iint\limits_D f(x,y) d\sigma = \lim_{n \to \infty} \displaystyle\sum_{i=1}^n \displaystyle\sum_{j=1}^n f(a+\frac{b-a}{n}i, c+\frac{d-c}{n}j) \cdot \frac{b-a}{n} \cdot \frac{d-c}{n} \\ \\ \\ 常用：&\int_0^1 \int_0^1 f(x,y) dxdy = \lim_{n \to \infty} \displaystyle\sum_{i=1}^n \displaystyle\sum_{j=1}^n f(\frac{i}{n}, \frac{j}{n})\cdot \frac{1}{n^2} \end{aligned} 常用：二重定积分精确定义：常用：​∫ab​f(x)dx=n→∞lim​i=1∑n​f(a+nb−a​i)nb−a​∫01​f(x)dx=n→∞lim​i=1∑n​f(ni​)⋅n1​∫0k​f(x)dx=n→∞lim​i=1∑kn​f(ni​)⋅n1​D∬​f(x,y)dσ=n→∞lim​i=1∑n​j=1∑n​f(a+nb−a​i,c+nd−c​j)⋅nb−a​⋅nd−c​∫01​∫01​f(x,y)dxdy=n→∞lim​i=1∑n​j=1∑n​f(ni​,nj​)⋅n21​​

∫ u d v = u v − ∫ v d u ∫ u v ( n + 1 ) d x = u v ( n ) − u ′ v ( n − 1 ) + u ′ ′ v ( n − 2 ) − . . . + ( − 1 ) n u ( n ) v + ( − 1 ) n + 1 ∫ u ( n + 1 ) v d x \begin{aligned} & \int u dv = uv - \int vdu \\ & \int uv^{(n+1)}dx = uv^{(n)} - u'v^{(n-1)} + u''v^{(n-2)} - ... + (-1)^nu^{(n)}v + (-1)^{n+1}\int u^{(n+1)}vdx \end{aligned} ​∫udv=uv−∫vdu∫uv(n+1)dx=uv(n)−u′v(n−1)+u′′v(n−2)−...+(−1)nu(n)v+(−1)n+1∫u(n+1)vdx​

∫ f ( x ) g ( x ) d x \int f(x) g(x) dx\\\\\\ ∫f(x)g(x)dx

f ( x ) f ′ ( x ) f ′ ′ ( x ) ⋯ f ( n ) ( x ) g ( x ) g 1 ( x ) g 2 ( x ) ⋯ g n ( x ) \def\arraystretch{2} \begin{array}{c:c:c:c:c} f(x) & f'(x) & f''(x) & \cdots & f^{(n)}(x) \\ \hline g(x) & g_1(x) & g_2(x) & \cdots & g_n(x) \\ \end{array} f(x)g(x)​f′(x)g1​(x)​f′′(x)g2​(x)​⋯⋯​f(n)(x)gn​(x)​​

∫ f ( x ) g ( x ) d x = f ( x ) g 1 ( x ) − f ′ ( x ) g 2 ( x ) + f ′ ′ ( x ) g 3 ( x ) − ⋯ ± ∫ f ( n ) ( x ) g n ( x ) d x 其 中 g 1 ( x ) 代 表 g ( x ) 的 积 分 ， 依 次 类 推 。 公 式 为 斜 对 角 线 加 减 交 替 。 \int f(x) g(x) dx = f(x)g_1(x) - f'(x)g_2(x) + f''(x)g_3(x) - \cdots \pm \int f^{(n)}(x) g_n(x) dx \\ \\ 其中 g_1(x) 代表g(x)的积分，依次类推。公式为斜对角线加减交替。 ∫f(x)g(x)dx=f(x)g1​(x)−f′(x)g2​(x)+f′′(x)g3​(x)−⋯±∫f(n)(x)gn​(x)dx其中g1​(x)代表g(x)的积分，依次类推。公式为斜对角线加减交替。

∫ 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 ∫ab​f(x)dx=∫ab​f(a+b−x)dx

∫ 0 π 2 sin ⁡ n x d x = ∫ 0 π 2 cos ⁡ n x d x = { n − 1 n ⋅ n − 3 n − 2 ⋅ ⋅ ⋅ ⋅ ⋅ 1 2 ⋅ π 2       n 为 正 偶 数 n − 1 n ⋅ n − 3 n − 2 ⋅ ⋅ ⋅ ⋅ ⋅ 2 3       n 为 大 于 1 的 奇 数 \int_0^\frac{\pi}{2} \sin ^n x dx = \int_0^\frac{\pi}{2} \cos ^n x dx = \begin{cases} \frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdot\cdot\cdot\cdot\cdot \frac{1}{2} \cdot \frac{\pi}{2} ~~~~~ n 为正偶数 \\ \frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdot\cdot\cdot\cdot\cdot \frac{2}{3} ~~~~~ n 为大于1的奇数 \\ \end{cases} ∫02π​​sinnxdx=∫02π​​cosnxdx={nn−1​⋅n−2n−3​⋅⋅⋅⋅⋅21​⋅2π​     n为正偶数nn−1​⋅n−2n−3​⋅⋅⋅⋅⋅32​     n为大于1的奇数​

∫ 1 + ∞ 1 x p d x    ⟹    { p > 1    ⟹    收 敛 p ≤ 1    ⟹    发 散 ∫ 0 1 1 x p d x    ⟹    { p < 1    ⟹    收 敛 p ≥ 1    ⟹    发 散 \begin{aligned} & \int_1^{+\infty} \frac{1}{x^p} dx \implies & \begin{cases} p>1 \implies 收敛 \\ p \le 1 \implies 发散 \end{cases} \\\\ & \int_0^1 \frac{1}{x^p} dx \implies & \begin{cases} p < 1 \implies 收敛 \\ p \ge 1 \implies 发散 \end{cases} \\\\ \end{aligned} ​∫1+∞​xp1​dx⟹∫01​xp1​dx⟹​{p>1⟹收敛p≤1⟹发散​{p 0 ∫ x α d x = 1 α + 1 x α + 1 + C       ( α ≠ − 1 ) ∫ 1 x d x = ln ⁡ ∣ x ∣ + C ∫ a x d x = a x ln ⁡ a + C       ( a > 0 , a ≠ 1 ) ∫ e x d x = e x + C ∫ sin ⁡ x d x = − cos ⁡ x + C ∫ cos ⁡ x d x = sin ⁡ x + C ∫ tan ⁡ x d x = − ln ⁡ ∣ cos ⁡ x ∣ + C ∫ cot ⁡ x d x = ln ⁡ ∣ sin ⁡ x ∣ + C ∫ sec ⁡ x d x = ln ⁡ ∣ sec ⁡ x + tan ⁡ x ∣ + C ∫ csc ⁡ x d x = ln ⁡ ∣ csc ⁡ x − cot ⁡ x ∣ + C ∫ sec ⁡ 2 x d x = tan ⁡ x + C ∫ csc ⁡ 2 x d x = − cot ⁡ x + C ∫ 1 a 2 + x 2 d x = 1 a arctan ⁡ x a + C ∫ 1 a 2 − x 2 d x = 1 2 a ln ⁡ ∣ a + x a − x ∣ + C ∫ 1 a 2 − x 2 d x = arcsin ⁡ x a + C ∫ 1 x 2 ± a 2 d x = ln ⁡ ∣ x + x 2 ± a 2 ∣ + C \begin{aligned} & 以下公式中，\alpha 与 a 均为常数，除声明者外，a>0 \\ \\ & \int x^\alpha dx = \frac{1}{\alpha +1} x^{\alpha + 1} + C ~~~~~(\alpha \ne -1) \\ \\ & \int \frac{1}{x} dx = \ln |x| + C \\ \\ & \int a^x dx = \frac{a^x}{\ln a} + C ~~~~~(a >0 , a\ne 1) \\ \\ & \int e^x dx = e^x + C \\ \\ & \int \sin x dx = -\cos x + C \\ \\ & \int \cos x dx = \sin x +C \\ \\ & \int \tan x dx = - \ln |\cos x| + C \\ \\ & \int \cot xdx = \ln |\sin x| + C \\ \\ & \int \sec x dx = \ln |\sec x + \tan x| + C \\ \\ & \int \csc x dx = \ln |\csc x - \cot x| + C \\ \\ & \int \sec ^2 x dx = \tan x + C \\ \\ & \int \csc^2x dx = - \cot x + C \\ \\ & \int \frac{1}{a^2 + x^2} dx = \frac{1}{a} \arctan \frac{x}{a} + C \\ \\ & \int \frac{1}{a^2-x^2} dx = \frac{1}{2a} \ln |\frac{a+x}{a-x}| + C \\ \\ & \int \frac{1}{\sqrt{a^2-x^2}} dx = \arcsin \frac{x}{a} + C \\ \\ & \int \frac{1}{\sqrt{x^2 \pm a^2}} dx = \ln |x+\sqrt{x^2 \pm a^2}| + C \end{aligned} ​以下公式中，α与a均为常数，除声明者外，a>0∫xαdx=α+11​xα+1+C     (α​=−1)∫x1​dx=ln∣x∣+C∫axdx=lnaax​+C     (a>0,a​=1)∫exdx=ex+C∫sinxdx=−cosx+C∫cosxdx=sinx+C∫tanxdx=−ln∣cosx∣+C∫cotxdx=ln∣sinx∣+C∫secxdx=ln∣secx+tanx∣+C∫cscxdx=ln∣cscx−cotx∣+C∫sec2xdx=tanx+C∫csc2xdx=−cotx+C∫a2+x21​dx=a1​arctanax​+C∫a2−x21​dx=2a1​ln∣a−xa+x​∣+C∫a2−x2 ​1​dx=arcsinax​+C∫x2±a2 ​1​dx=ln∣x+x2±a2 ​∣+C​

∫ − ∞ + ∞ e − x 2 d x = 2 ∫ 0 + ∞ e − x 2 d x = π ∫ 0 + ∞ x n e − x d x = n ! ∫ − a a f ( x ) d x = ∫ 0 a [ f ( x ) + f ( − x ) ] d x ∫ 0 π x f ( sin ⁡ x ) d x = π 2 ∫ 0 π f ( sin ⁡ x ) d x = π ∫ 0 π 2 f ( sin ⁡ x ) d x ∫ a b f ( x ) d x = ( b − a ) ∫ 0 1 f [ a + ( b − a ) x ] d x \begin{aligned} & \int_{-\infty}^{+\infty} e^{-x^2} dx = 2\int_{0}^{+\infty} e^{-x^2} dx = \sqrt{\pi} \\ \\ & \int_{0}^{+\infty} x^n e^{-x} dx = n! \\ \\ & \int_{-a}^{a} f(x) dx = \int_0^a [f(x)+f(-x)]dx \\ \\ & \int_0^\pi xf(\sin x) dx = \frac{\pi}{2} \int_0^\pi f(\sin x)dx = \pi \int_0^{\frac{\pi}{2}} f(\sin x) dx \\ \\ & \int_a^b f(x) dx = (b-a) \int_0^1 f[a+(b-a)x] dx \end{aligned} ​∫−∞+∞​e−x2dx=2∫0+∞​e−x2dx=π ​∫0+∞​xne−xdx=n!∫−aa​f(x)dx=∫0a​[f(x)+f(−x)]dx∫0π​xf(sinx)dx=2π​∫0π​f(sinx)dx=π∫02π​​f(sinx)dx∫ab​f(x)dx=(b−a)∫01​f[a+(b−a)x]dx​

P n ( x ) ( a 1 x + b 1 ) ( a 2 x + b 2 ) ( a 3 x + b 3 ) = A 1 a 1 x + b 1 + A 2 a 2 x + b 2 + A 3 a 2 x + b 2 P n ( x ) Q m ( x ) ( a x + b ) 3 = A ( x ) Q m ( x ) + A 1 ( a x + b ) 3 + A 2 ( a x + b ) 2 + A 3 ( a x + b ) P n ( x ) Q m ( x ) ( a x 2 + b x + c ) 3 = A ( x ) Q m ( x ) + A 1 x + B 1 ( a x 2 + b x + c ) 3 + A 2 x + B 2 ( a x 2 + b x + c ) 2 + A 3 x + B 3 ( a x 2 + b x + c ) \begin{aligned} & \frac{P_n(x)}{(a_1 x+b_1)(a_2 x+b_2)(a_3 x+b_3)} = \frac{A_1}{a_1 x+b_1}+\frac{A_2}{a_2 x+b_2}+\frac{A_3}{a_2 x+b_2} \\ \\ & \frac{P_n(x)}{Q_m(x)(ax+b)^3} = \frac{A(x)}{Q_m(x)} + \frac{A_1}{(ax+b)^3} + \frac{A_2}{(ax+b)^2} + \frac{A_3}{(ax+b)} \\ \\ & \frac{P_n(x)}{Q_m(x)(ax^2+bx+c)^3} = \frac{A(x)}{Q_m(x)} + \frac{A_1x+B_1}{(ax^2+bx+c)^3} + \frac{A_2x+B_2}{(ax^2+bx+c)^2} + \frac{A_3x+B_3}{(ax^2+bx+c)} \\ \\ \end{aligned} ​(a1​x+b1​)(a2​x+b2​)(a3​x+b3​)Pn​(x)​=a1​x+b1​A1​​+a2​x+b2​A2​​+a2​x+b2​A3​​Qm​(x)(ax+b)3Pn​(x)​=Qm​(x)A(x)​+(ax+b)3A1​​+(ax+b)2A2​​+(ax+b)A3​​Qm​(x)(ax2+bx+c)3Pn​(x)​=Qm​(x)A(x)​+(ax2+bx+c)3A1​x+B1​​+(ax2+bx+c)2A2​x+B2​​+(ax2+bx+c)A3​x+B3​​​

f ( x ) 在 [ a , b ] 上 的 平 均 值 为 ： ∫ a b f ( x ) d x b − a f(x) 在[a,b]上的平均值为： \frac{\int_a^b f(x) dx}{b-a} f(x)在[a,b]上的平均值为：b−a∫ab​f(x)dx​

y = y 1 ( x ) 与 y = y 2 ( x ) ， 及 x = a ， x = b （ a < b ） 所 围 成 的 平 面 图 形 面 积 ： S = ∫ a b ∣ y 1 ( x ) − y 2 ( x ) ∣ d x 曲 线 r = r 1 ( θ ) 与 r = r 2 ( θ ) 与 两 射 线 θ = α 与 θ = β （ 0 < β − α ≤ 2 π ） 所 围 成 的 曲 边 扇 形 的 面 积 ： S = 1 2 ∫ α β ∣ r 1 2 ( θ ) − r 2 2 ( θ ) ∣ d θ \begin{aligned} & y=y_1(x) 与 y=y_2(x)，及x=a，x=b（a