【408直通车】（考研数一、二、三合集）高等数学公式全覆盖（上）

数学集合定义总结：

自然数集（ N \mathbb{N} N） ：包括0、1、2、3等正整数，即 { 0 , 1 , 2 , ... } \{0, 1, 2, \ldots\} {0,1,2,...}。
整数集（ Z \mathbb{Z} Z） ：包括负整数、0和正整数，即 { ... , − 1 , 0 , 1 , ... } \{\ldots, -1, 0, 1, \ldots\} {...,−1,0,1,...}。
非零自然数集（ N ∗ \mathbb{N^*} N∗或 N + \mathbb{N+} N+） ：也称为正整数集，包括1、2、3等正整数，即 { 1 , 2 , 3 , ... } \{1, 2, 3, \ldots\} {1,2,3,...}。
有理数集（ Q \mathbb{Q} Q） ：包括所有可以表示为两个整数之比的数，形如 a b \frac{a}{b} ba，其中 a a a 和 b b b 是整数且 b ≠ 0 b \neq 0 b=0。
无理数集（ R ∖ Q \mathbb{R} \setminus \mathbb{Q} R∖Q） ：包括不能被有理数表示的实数，如 2 \sqrt{2} 2 、 π \pi π 等。
实数集（ R \mathbb{R} R）：包括所有有理数和无理数。
复数集（ C \mathbb{C} C） ：包括实部和虚部均为实数的数，形如 a + b i a + bi a+bi，其中 a a a 和 b b b 是实数，且 i i i 是虚数单位。
函数集合（ N N N^N NN） ：包含所有从自然数到自然数的可能函数，表示为 { f : N → N } \{f: \mathbb{N} \rightarrow \mathbb{N}\} {f:N→N}。

代数

x 3 − 1 = ( x − 1 ) ( x 2 + x + 1 ) x^3 - 1 = (x - 1)(x^2 + x + 1) x3−1=(x−1)(x2+x+1)
1 2 + 2 2 + 3 2 + ⋯ + n 2 = 1 6 n ( n + 1 ) ( 2 n + 1 ) \frac{1}{2} + 2^2 + 3^2 + \cdots + n^2 = \frac{1}{6}n(n + 1)(2n + 1) 21+22+32+⋯+n2=61n(n+1)(2n+1)
a 3 ± b 3 = ( a ± b ) ( a 2 ∓ a b + b 2 ) a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2) a3±b3=(a±b)(a2∓ab+b2)
a n − b n = ( a − b ) ( a n − 1 + a n − 2 b + a n − 3 b 2 + ⋯ + a b n − 2 + b n − 1 ) a^n - b^n = (a - b)(a^{n-1} + a^{n-2}b + a^{n-3}b^2 + \cdots + ab^{n-2} + b^{n-1}) an−bn=(a−b)(an−1+an−2b+an−3b2+⋯+abn−2+bn−1)

排列组合

排列公式
- A m n = n ( n − 1 ) ⋯ ( n − m + 1 ) = n ! ( n − m ) ! A_m^n = n(n - 1)\cdots(n - m + 1) = \frac{n!}{(n - m)!} Amn=n(n−1)⋯(n−m+1)=(n−m)!n!
  (其中 n , m ∈ N ∗ n, m \in \mathbb{N^*} n,m∈N∗ 且 m ≤ n m \leq n m≤n)
组合公式
- C m n = ( n m ) = A m n m ! = n ( n − 1 ) ⋯ ( n − m + 1 ) m ! = n ! m ! ( n − m ) ! C_m^n = \binom{n}{m} = \frac{A_m^n}{m!} = \frac{n(n - 1)\cdots(n - m + 1)}{m!} = \frac{n!}{m!(n - m)!} Cmn=(mn)=m!Amn=m!n(n−1)⋯(n−m+1)=m!(n−m)!n!
  (其中 n ∈ N N , m ∈ N n \in \mathbb{N^N}, m \in \mathbb{N} n∈NN,m∈N 且 m ≤ n m \leq n m≤n)
组合公式性质
- C m n + C m − 1 n = C m n + 1 C_m^n + C_{m-1}^n = C_{m}^{n+1} Cmn+Cm−1n=Cmn+1

等差等比求和

前 n 项和公式（等差数列）：
- S n = n ( a 1 + a n ) 2 = n a 1 + n ( n − 1 ) 2 d = d 2 n 2 + ( a 1 − 1 2 d ) n S_n = \frac{n(a_1 + a_n)}{2} = na_1 + \frac{n(n - 1)}{2}d=\frac{d}{2}n^2 + (a_1 - \frac{1}{2}d)n Sn=2n(a1+an)=na1+2n(n−1)d=2dn2+(a1−21d)n
前 n 项和公式（等比数列）：
- S n = { a 1 ( 1 − q n ) 1 − q , q ≠ 1 n a 1 , q = 1 S_n = \begin{cases} \frac{a_1(1-q^n)}{1-q}, & q \neq 1 \\ na_1, & q = 1 \end{cases} Sn={1−qa1(1−qn),na1,q=1q=1

定积分

∫ a b f ( x ) d x = lim ⁡ λ → 0 ∑ i = 1 n f ( ξ i ) Δ x i \int_a^b f(x)dx = \lim\limits_{\lambda \to 0} \sum_{i=1}^{n} f(\xi_i)\Delta x_i ∫abf(x)dx=λ→0lim∑i=1nf(ξi)Δxi

三角函数

和角与差角公式

sin ⁡ ( α ± β ) = sin ⁡ α cos ⁡ β ± cos ⁡ α sin ⁡ β \sin(\alpha \pm \beta) = \sin\alpha \cos\beta \pm \cos\alpha \sin\beta sin(α±β)=sinαcosβ±cosαsinβ
cos ⁡ ( α ± β ) = cos ⁡ α cos ⁡ β ∓ sin ⁡ α sin ⁡ β \cos(\alpha \pm \beta) = \cos\alpha \cos\beta \mp \sin\alpha \sin\beta cos(α±β)=cosαcosβ∓sinαsinβ
tan ⁡ ( α ± β ) = tan ⁡ α ± tan ⁡ β 1 ∓ tan ⁡ α tan ⁡ β \tan(\alpha \pm \beta) = \frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha \tan\beta} tan(α±β)=1∓tanαtanβtanα±tanβ
sin ⁡ ( α + β ) sin ⁡ ( α − β ) = sin ⁡ 2 α − sin ⁡ 2 β \sin(\alpha + \beta) \sin(\alpha - \beta) = \sin^2 \alpha - \sin^2 \beta sin(α+β)sin(α−β)=sin2α−sin2β (平方正弦公式)
cos ⁡ ( α + β ) cos ⁡ ( α − β ) = cos ⁡ 2 α − sin ⁡ 2 β \cos(\alpha + \beta) \cos(\alpha - \beta) = \cos^2 \alpha - \sin^2 \beta cos(α+β)cos(α−β)=cos2α−sin2β
a sin ⁡ α + b cos ⁡ α = a 2 + b 2 sin ⁡ ( α + ϕ ) a \sin \alpha + b \cos \alpha = \sqrt{a^2 + b^2} \sin(\alpha + \phi) asinα+bcosα=a2+b2 sin(α+ϕ) (辅助角 ϕ \phi ϕ所在象限由点 ( a , b ) (a, b) (a,b)的象限决定， tan ⁡ ϕ = b a \tan \phi = \frac{b}{a} tanϕ=ab)

倍角公式

sin ⁡ ( 2 α ) = 2 sin ⁡ ( α ) cos ⁡ ( α ) \sin(2\alpha) = 2\sin(\alpha)\cos(\alpha) sin(2α)=2sin(α)cos(α)
cos ⁡ ( 2 α ) = 2 cos ⁡ 2 ( α ) − 1 = 1 − 2 sin ⁡ 2 ( α ) = cos ⁡ 2 ( α ) − sin ⁡ 2 ( α ) \cos(2\alpha) = 2\cos^2(\alpha) - 1 = 1 - 2\sin^2(\alpha) = \cos^2(\alpha) - \sin^2(\alpha) cos(2α)=2cos2(α)−1=1−2sin2(α)=cos2(α)−sin2(α)
tan ⁡ ( 2 α ) = 2 tan ⁡ ( α ) 1 − tan ⁡ 2 ( α ) \tan(2\alpha) = \frac{2\tan(\alpha)}{1 - \tan^2(\alpha)} tan(2α)=1−tan2(α)2tan(α)

半角公式

sin ⁡ ( α 2 ) = ± 1 − cos ⁡ α 2 \sin\left(\frac{\alpha}{2}\right) = \pm \sqrt{\frac{1 - \cos \alpha}{2}} sin(2α)=±21−cosα
cos ⁡ ( α 2 ) = ± 1 + cos ⁡ α 2 \cos\left(\frac{\alpha}{2}\right) = \pm \sqrt{\frac{1 + \cos \alpha}{2}} cos(2α)=±21+cosα
tan ⁡ ( α 2 ) = ± 1 − cos ⁡ α 1 + cos ⁡ α = 1 − cos ⁡ α sin ⁡ α = sin ⁡ α 1 + cos ⁡ α \tan\left(\frac{\alpha}{2}\right) = \pm \sqrt{\frac{1 - \cos \alpha}{1 + \cos \alpha}} = \frac{1 - \cos \alpha}{\sin \alpha} = \frac{\sin \alpha}{1 + \cos \alpha} tan(2α)=±1+cosα1−cosα =sinα1−cosα=1+cosαsinα

和差化积与积化和差

口诀:

正加正，正在前，余加余，余并肩，

正减正，余在前，余减余，负正弦.

1. 和差化积：

sin ⁡ α + sin ⁡ β = 2 sin ⁡ ( α + β 2 ) cos ⁡ ( α − β 2 ) \sin\alpha + \sin\beta = 2\sin\left(\frac{\alpha + \beta}{2}\right)\cos\left(\frac{\alpha - \beta}{2}\right) sinα+sinβ=2sin(2α+β)cos(2α−β)
sin ⁡ α − sin ⁡ β = 2 cos ⁡ ( α + β 2 ) sin ⁡ ( α − β 2 ) \sin\alpha - \sin\beta = 2\cos\left(\frac{\alpha + \beta}{2}\right)\sin\left(\frac{\alpha - \beta}{2}\right) sinα−sinβ=2cos(2α+β)sin(2α−β)
cos ⁡ α + cos ⁡ β = 2 cos ⁡ ( α + β 2 ) cos ⁡ ( α − β 2 ) \cos\alpha + \cos\beta = 2\cos\left(\frac{\alpha + \beta}{2}\right)\cos\left(\frac{\alpha - \beta}{2}\right) cosα+cosβ=2cos(2α+β)cos(2α−β)
cos ⁡ α − cos ⁡ β = − 2 sin ⁡ ( α + β 2 ) sin ⁡ ( α − β 2 ) \cos\alpha - \cos\beta = -2\sin\left(\frac{\alpha + \beta}{2}\right)\sin\left(\frac{\alpha - \beta}{2}\right) cosα−cosβ=−2sin(2α+β)sin(2α−β)
tan ⁡ A + tan ⁡ B = sin ⁡ ( A + B ) cos ⁡ A cos ⁡ B = tan ⁡ ( A + B ) ( 1 − tan ⁡ A tan ⁡ B ) \tan A + \tan B = \frac{\sin(A+B)}{\cos A \cos B} = \tan(A+B)(1-\tan A \tan B) tanA+tanB=cosAcosBsin(A+B)=tan(A+B)(1−tanAtanB)
tan ⁡ A − tan ⁡ B = sin ⁡ ( A − B ) cos ⁡ A cos ⁡ B = tan ⁡ ( A − B ) ( 1 + tan ⁡ A tan ⁡ B ) \tan A - \tan B = \frac{\sin(A-B)}{\cos A \cos B} = \tan(A-B)(1+\tan A \tan B) tanA−tanB=cosAcosBsin(A−B)=tan(A−B)(1+tanAtanB)

2. 积化和差公式：

cos ⁡ a sin ⁡ β = 1 2 [ sin ⁡ ( a + β ) − sin ⁡ ( a − β ) ] \cos a \sin \beta = \frac{1}{2} [\sin(a + \beta) - \sin(a - \beta)] cosasinβ=21[sin(a+β)−sin(a−β)]
sin ⁡ a cos ⁡ β = 1 2 [ sin ⁡ ( a + β ) + sin ⁡ ( a − β ) ] \sin a \cos \beta = \frac{1}{2} [\sin (a + \beta) + \sin(a - \beta)] sinacosβ=21[sin(a+β)+sin(a−β)]
cos ⁡ a cos ⁡ β = 1 2 [ cos ⁡ ( a + β ) + cos ⁡ ( a − β ) ] \cos a \cos \beta = \frac{1}{2} [\cos(a + \beta) + \cos(a - \beta)] cosacosβ=21[cos(a+β)+cos(a−β)]
sin ⁡ a sin ⁡ β = − 1 2 [ cos ⁡ ( a + β ) − cos ⁡ ( a − β ) ] \sin a \sin \beta = -\frac{1}{2} [\cos(a + \beta) - \cos(a - \beta)] sinasinβ=−21[cos(a+β)−cos(a−β)]

三角和

sin ⁡ ( α + β + γ ) = sin ⁡ α ⋅ cos ⁡ β ⋅ cos ⁡ γ + cos ⁡ α ⋅ sin ⁡ β ⋅ cos ⁡ γ + cos ⁡ α ⋅ cos ⁡ β ⋅ sin ⁡ γ − sin ⁡ α ⋅ sin ⁡ β ⋅ sin ⁡ γ \sin(\alpha + \beta + \gamma) = \sin\alpha \cdot \cos\beta \cdot \cos\gamma + \cos\alpha \cdot \sin\beta \cdot \cos\gamma + \cos\alpha \cdot \cos\beta \cdot \sin\gamma - \sin\alpha \cdot \sin\beta \cdot \sin\gamma sin(α+β+γ)=sinα⋅cosβ⋅cosγ+cosα⋅sinβ⋅cosγ+cosα⋅cosβ⋅sinγ−sinα⋅sinβ⋅sinγ
cos ⁡ ( α + β + γ ) = cos ⁡ α ⋅ cos ⁡ β ⋅ cos ⁡ γ − cos ⁡ α ⋅ sin ⁡ β ⋅ sin ⁡ γ − sin ⁡ α ⋅ cos ⁡ β ⋅ sin ⁡ γ − sin ⁡ α ⋅ sin ⁡ β ⋅ cos ⁡ γ \cos(\alpha + \beta + \gamma) = \cos\alpha \cdot \cos\beta \cdot \cos\gamma - \cos\alpha \cdot \sin\beta \cdot \sin\gamma - \sin\alpha \cdot \cos\beta \cdot \sin\gamma - \sin\alpha \cdot \sin\beta \cdot \cos\gamma cos(α+β+γ)=cosα⋅cosβ⋅cosγ−cosα⋅sinβ⋅sinγ−sinα⋅cosβ⋅sinγ−sinα⋅sinβ⋅cosγ
tan ⁡ ( α + β + γ ) = tan ⁡ α + tan ⁡ β + tan ⁡ γ − tan ⁡ α ⋅ tan ⁡ β ⋅ tan ⁡ γ 1 − tan ⁡ α ⋅ tan ⁡ β − tan ⁡ β ⋅ tan ⁡ γ − tan ⁡ γ ⋅ tan ⁡ α \tan(\alpha + \beta + \gamma) = \dfrac{\tan\alpha + \tan\beta + \tan\gamma - \tan\alpha \cdot \tan\beta \cdot \tan\gamma}{1 - \tan\alpha \cdot \tan\beta - \tan\beta \cdot \tan\gamma - \tan\gamma \cdot \tan\alpha} tan(α+β+γ)=1−tanα⋅tanβ−tanβ⋅tanγ−tanγ⋅tanαtanα+tanβ+tanγ−tanα⋅tanβ⋅tanγ

诱导公式：

sin ⁡ ( − α ) = − sin ⁡ α \sin(-\alpha) = -\sin\alpha sin(−α)=−sinα
cos ⁡ ( − α ) = cos ⁡ α \cos(-\alpha) = \cos\alpha cos(−α)=cosα
tan ⁡ ( − α ) = − tan ⁡ α \tan(-\alpha) = -\tan\alpha tan(−α)=−tanα
sin ⁡ ( π 2 − α ) = cos ⁡ α \sin\left(\frac{\pi}{2}-\alpha\right) = \cos\alpha sin(2π−α)=cosα
cos ⁡ ( π 2 − α ) = sin ⁡ α \cos\left(\frac{\pi}{2}-\alpha\right) = \sin\alpha cos(2π−α)=sinα
sin ⁡ ( π 2 + α ) = cos ⁡ α \sin\left(\frac{\pi}{2}+\alpha\right) = \cos\alpha sin(2π+α)=cosα
cos ⁡ ( π 2 + α ) = − sin ⁡ α \cos\left(\frac{\pi}{2}+\alpha\right) = -\sin\alpha cos(2π+α)=−sinα
sin ⁡ ( π − α ) = sin ⁡ α \sin(\pi-\alpha) = \sin\alpha sin(π−α)=sinα
cos ⁡ ( π − α ) = − cos ⁡ α \cos(\pi-\alpha) = -\cos\alpha cos(π−α)=−cosα
sin ⁡ ( π + α ) = − sin ⁡ α \sin(\pi+\alpha) = -\sin\alpha sin(π+α)=−sinα
cos ⁡ ( π + α ) = − cos ⁡ α \cos(\pi+\alpha) = -\cos\alpha cos(π+α)=−cosα

不等式：

a + b ≥ 2 a b , a > 0 , b > 0 a + b \geq 2\sqrt{ab}, a > 0, b > 0 a+b≥2ab ,a>0,b>0
a b ≤ ( a + b ) 2 4 ab \leq \frac{(a+b)^2}{4} ab≤4(a+b)2
− ∣ a ∣ ≤ a ≤ ∣ a ∣ -|a| \leq a \leq |a| −∣a∣≤a≤∣a∣
∣ ∣ a ∣ − ∣ b ∣ ∣ ≤ ∣ a ± b ∣ ≤ ∣ a ∣ + ∣ b ∣ ||a| - |b|| \leq |a \pm b| \leq |a| + |b| ∣∣a∣−∣b∣∣≤∣a±b∣≤∣a∣+∣b∣
a − 1 < [ a ] ≤ a , [ a ] a - 1 < [a] \leq a, [a] a−1<[a]≤a,[a] 表示对 a a a取整，即不超过 a a a的最大整数
对于 x > 0 , x < sin ⁡ x x > 0, x < \sin x x>0,x<sinx；对于 x < 0 , x > sin ⁡ x x < 0, x > \sin x x<0,x>sinx
arctan ⁡ x < sin ⁡ x < x < arcsin ⁡ x < tan ⁡ x \arctan x < \sin x < x < \arcsin x < \tan x arctanx<sinx<x<arcsinx<tanx，其中 0 < x < π 2 0 < x < \frac{\pi}{2} 0<x<2π
对于 x > 0 , x > ln ⁡ ( 1 + x ) x > 0, x > \ln(1 + x) x>0,x>ln(1+x)
e x − 1 ≥ x , x ∈ R e^x - 1 \geq x, x \in \mathbb{R} ex−1≥x,x∈R

函数关系和微分公式：

[ sin ⁡ ( a x + b ) ] ( n ) = a n sin ⁡ ( a x + b + n π 2 ) [ \sin(ax + b) ]^{(n)} = a^n \sin(ax + b+ \frac{n\pi}{2}) [sin(ax+b)](n)=ansin(ax+b+2nπ)
[ cos ⁡ ( a x + b ) ] ( n ) = a n cos ⁡ ( a x + b + n π 2 ) [ \cos(ax + b) ]^{(n)} = a^n \cos(ax + b+\frac{n\pi}{2}) [cos(ax+b)](n)=ancos(ax+b+2nπ)
[ ln ⁡ ( a x + b ) ] ( n ) = ( − 1 ) n − 1 ( n − 1 ) ! a n ( a x + b ) n [ \ln(ax + b) ]^{(n)} = \frac{(-1)^{n-1}(n-1)!a^n}{(ax+b)^n} [ln(ax+b)](n)=(ax+b)n(−1)n−1(n−1)!an
[ ( a x + b ) a ] ( n ) = { α ( α − 1 ) ⋯ ( α − n + 1 ) ( a x + b ) a − n a n , n < α n ! , n = α 0 , n > α [ (ax + b)^{a} ](n) = \begin{cases} \alpha(\alpha - 1) \cdots (\alpha - n + 1)(ax + b)^{a-n} a^n, & n < \alpha \\ n! , & n = \alpha \\ 0 , & n > \alpha \end{cases} [(ax+b)a](n)=⎩ ⎨ ⎧α(α−1)⋯(α−n+1)(ax+b)a−nan,n!,0,n<αn=αn>α
Gamma 函数:
- Γ ( z ) = ∫ 0 ∞ e − t t z − 1 d t \Gamma(z) = \int_{0}^{\infty} e^{-t} t^{z-1} dt Γ(z)=∫0∞e−ttz−1dt
- Γ ( z + 1 ) = z Γ ( z ) \Gamma(z + 1) = z\Gamma(z) Γ(z+1)=zΓ(z)
- Γ ( 1 2 ) = π \Gamma\left(\frac{1}{2}\right) = \sqrt{\pi} Γ(21)=π
微分公式:
- ( c ) ′ = 0 (c)' = 0 (c)′=0
- y = x α y = x^{\alpha} y=xα ( α \alpha α 为实数), y ′ = α x α − 1 y' = \alpha x^{\alpha-1} y′=αxα−1
- ( a x ) ′ = a x ln ⁡ a (a^x)' = a^x \ln a (ax)′=axlna, ( e x ) ′ = e x ( log ⁡ a x ) ′ = 1 x ln ⁡ a (e^x)' = e^x (\log_a x)' = \frac{1}{x \ln a} (ex)′=ex(logax)′=xlna1
- ( ln ⁡ ∣ x ∣ ) ′ = 1 x (\ln |x|)' = \frac{1}{x} (ln∣x∣)′=x1,
- ( sin ⁡ x ) ′ = cos ⁡ x (\sin x)' = \cos x (sinx)′=cosx, ( cos ⁡ x ) ′ = − sin ⁡ x (\cos x)' = -\sin x (cosx)′=−sinx
- ( tan ⁡ x ) ′ = sec ⁡ 2 x (\tan x)' = \sec^2 x (tanx)′=sec2x, ( cot ⁡ x ) ′ = − csc ⁡ 2 x (\cot x)' = -\csc^2 x (cotx)′=−csc2x
- ( sec ⁡ x ) ′ = sec ⁡ x tan ⁡ x (\sec x)' = \sec x \tan x (secx)′=secxtanx, ( csc ⁡ x ) ′ = − csc ⁡ x cot ⁡ x (\csc x)' = -\csc x \cot x (cscx)′=−cscxcotx
- ( arcsin ⁡ x ) ′ = 1 1 − x 2 (\arcsin x)' = \frac{1}{\sqrt{1-x^2}} (arcsinx)′=1−x2 1, ( arccos ⁡ x ) ′ = − 1 1 − x 2 (\arccos x)' = -\frac{1}{\sqrt{1-x^2}} (arccosx)′=−1−x2 1
- ( arctan ⁡ x ) ′ = 1 1 + x 2 (\arctan x)' = \frac{1}{1+x^2} (arctanx)′=1+x21, ( arccot x ) ′ = − 1 1 + x 2 (\text{arccot } x)' = -\frac{1}{1+x^2} (arccot x)′=−1+x21
- ( u v ) ( n ) = ∑ λ = 0 n C n k u ( k ) v ( n − k ) (uv)^{(n)} = \sum_{\lambda=0}^{n} C_{n}^{k} u^{(k)} v^{(n-k)} (uv)(n)=∑λ=0nCnku(k)v(n−k)
- ( a x ) ( n ) = a x ln ⁡ n a (a^x)^{(n)} = a^x \ln^n a (ax)(n)=axlnna ( a > 0 a > 0 a>0)
- ( e x ) ( n ) = e x (e^x)^{(n)} = e^x (ex)(n)=ex
- ( sin ⁡ k x ) ( n ) = k n sin ⁡ ( k x + n ⋅ π 2 ) (\sin kx)^{(n)} = k^n \sin\left(kx + n\cdot\frac{\pi}{2}\right) (sinkx)(n)=knsin(kx+n⋅2π)
- ( cos ⁡ k x ) ( n ) = k n cos ⁡ ( k x + n ⋅ π 2 ) (\cos kx)^{(n)} = k^n \cos\left(kx + n\cdot \frac{\pi}{2}\right) (coskx)(n)=kncos(kx+n⋅2π)
- ( x m ) ( n ) = m ( m − 1 ) ⋯ ( m − n + 1 ) x m − n (x^m)^{(n)} = m(m-1) \cdots (m-n+1) x^{m-n} (xm)(n)=m(m−1)⋯(m−n+1)xm−n
- ( ln ⁡ x ) ( n ) = ( − 1 ) n − 1 ( n − 1 ) ! x n (\ln x)^{(n)} = (-1)^{n-1} \frac{(n-1)! }{x^n} (lnx)(n)=(−1)n−1xn(n−1)!
- ( 1 1 + x ) ( n ) = ( − 1 ) n ⋅ n ! ⋅ ( x + 1 ) − ( n + 1 ) \left(\frac{1}{1+x}\right)^{(n)} = (-1)^n \cdot n! \cdot (x+1)^{-(n+1)} (1+x1)(n)=(−1)n⋅n!⋅(x+1)−(n+1)
- ( 1 a x + b ) ( n ) = ( − 1 ) n ⋅ n ! ⋅ ( a x + b ) − ( n + 1 ) ⋅ a n \left(\frac{1}{ax+b}\right)^{(n)} = (-1)^n \cdot n! \cdot(ax + b)^{-(n+1)} \cdot a^n (ax+b1)(n)=(−1)n⋅n!⋅(ax+b)−(n+1)⋅an
- ln ⁡ ′ ( x + 1 + x 2 ) = 1 1 + x 2 \ln'(x+\sqrt{1 + x^2}) = \frac{1}{\sqrt{1+x^2}} ln′(x+1+x2 )=1+x2 1
- ln ⁡ ′ ( sec ⁡ x + tan ⁡ x ) = 1 cos ⁡ x \ln'(\sec x + \tan x) = \frac{1}{\cos x} ln′(secx+tanx)=cosx1

积分公式：

∫ x k d x = 1 k + 1 x k + 1 + C ( k ≠ − 1 ) \int x^k \, dx = \frac{1}{k+1}x^{k+1} + C\quad (k \neq -1) ∫xkdx=k+11xk+1+C(k=−1)
∫ 1 x 2 d x = − 1 x + C \int \frac{1}{x^2} \, dx = -\frac{1}{x} + C ∫x21dx=−x1+C
∫ 1 x d x = 2 x + C \int \frac{1}{\sqrt{x}} \, dx = 2\sqrt{x} + C ∫x 1dx=2x +C
∫ 1 x d x = ln ⁡ ∣ x ∣ + C \int \frac{1}{x} \, dx = \ln |x| + C ∫x1dx=ln∣x∣+C
∫ a x d x = a x ln ⁡ a + C ( a > 0 , a ≠ 1 ) \int a^x \, dx = \frac{a^x}{\ln a} + C \quad (a > 0, a \neq 1) ∫axdx=lnaax+C(a>0,a=1)
∫ e x d x = e x + C \int e^x \, dx = e^x + C ∫exdx=ex+C

三角
- ∫ cos ⁡ x d x = sin ⁡ x + C \int \cos x \, dx = \sin x + C ∫cosxdx=sinx+C
- ∫ sin ⁡ x d x = − cos ⁡ x + C \int \sin x \, dx = -\cos x + C ∫sinxdx=−cosx+C
- ∫ 1 cos ⁡ 2 x d x = ∫ sec ⁡ 2 x d x = tan ⁡ x + C \int \frac{1}{\cos^2 x} \, dx = \int \sec^2 x \, dx = \tan x + C ∫cos2x1dx=∫sec2xdx=tanx+C
- ∫ 1 sin ⁡ 2 x d x = ∫ csc ⁡ 2 x d x = − cot ⁡ x + C \int \frac{1}{\sin^2 x} \, dx = \int \csc^2 x \, dx = -\cot x + C ∫sin2x1dx=∫csc2xdx=−cotx+C
- ∫ sec ⁡ x tan ⁡ x d x = sec ⁡ x + C \int \sec x \tan x \, dx = \sec x + C ∫secxtanxdx=secx+C
- ∫ csc ⁡ x cot ⁡ x d x = − csc ⁡ x + C \int \csc x \cot x \, dx = -\csc x + C ∫cscxcotxdx=−cscx+C
arc
- ∫ d x a 2 + x 2 = 1 a arctan ⁡ ( x a ) + C \int \frac{dx}{a^2+x^2} = \frac{1}{a}\arctan\left(\frac{x}{a}\right) + C ∫a2+x2dx=a1arctan(ax)+C
  - ∫ d x 1 + x 2 = arctan ⁡ x + C \int \frac{dx}{1+x^2} = \arctan x + C ∫1+x2dx=arctanx+C
- ∫ d x a 2 − x 2 = arcsin ⁡ ( x a ) + C \int \frac{dx}{\sqrt{{a^2-x^2}}} = \arcsin\left(\frac{x}{a}\right) + C ∫a2−x2 dx=arcsin(ax)+C
- ∫ d x 1 − x 2 = arcsin ⁡ x + C \int \frac{dx}{\sqrt{{1-x^2}}} = \arcsin x + C ∫1−x2 dx=arcsinx+C
ln
- ∫ d x x 2 − a 2 = 1 2 a ln ⁡ ∣ a − x a + x ∣ + C \int \frac{dx}{x^2-a^2} = \frac{1}{2a}\ln \left|\frac{a-x}{a+x}\right| + C ∫x2−a2dx=2a1ln a+xa−x +C
- ∫ d x a 2 − x 2 = 1 2 a ln ⁡ ∣ a + x a − x ∣ + C \int \frac{dx}{a^2-x^2} = \frac{1}{2a}\ln \left|\frac{a+x}{a-x}\right| + C ∫a2−x2dx=2a1ln a−xa+x +C
  - ∫ d x 1 − x 2 = 1 2 ln ⁡ ∣ 1 + x 1 − x ∣ + C \int \frac{dx}{1-x^2} = \frac{1}{2}\ln\left| \frac{1+x}{1-x} \right| + C ∫1−x2dx=21ln 1−x1+x +C
- ∫ d x x 2 + a 2 = ln ⁡ ( x + x 2 + a 2 ) + C \int \frac{dx}{\sqrt{{x^2 + a^2}}} = \ln ( x + \sqrt{x^2+ a^2} ) + C ∫x2+a2 dx=ln(x+x2+a2 )+C
- ∫ d x x 2 − a 2 = ln ⁡ ∣ x + x 2 − a 2 ∣ + C ( ∣ x ∣ > ∣ a ∣ ) \int \frac{dx}{\sqrt{{x^2 - a^2}}} = \ln | x + \sqrt{x^2 - a^2} | + C \quad (|x| > |a|) ∫x2−a2 dx=ln∣x+x2−a2 ∣+C(∣x∣>∣a∣)
- ∫ tan ⁡ x d x = − ln ⁡ ∣ cos ⁡ x ∣ + C \int \tan x \, dx = -\ln |\cos x| + C ∫tanxdx=−ln∣cosx∣+C
- ∫ sec ⁡ x d x = ∫ 1 cos ⁡ x d x = ln ⁡ ∣ tan ⁡ x + sec ⁡ x ∣ + C \int \sec x \, dx = \int \frac{1}{\cos x} \, dx = \ln |\tan{x} + \sec{x}| + C ∫secxdx=∫cosx1dx=ln∣tanx+secx∣+C
- ∫ cot ⁡ x d x = ln ⁡ ∣ sin ⁡ x ∣ + C \int \cot x \, dx = \ln |\sin x| + C ∫cotxdx=ln∣sinx∣+C
- ∫ csc ⁡ x d x = ∫ 1 sin ⁡ x d x = ln ⁡ ∣ csc ⁡ x − cot ⁡ x ∣ + C = ln ⁡ ∣ tan ⁡ x 2 ∣ + C \int \csc x \, dx = \int \frac{1}{\sin x} \, dx = \ln |\csc{x} - \cot{x}| + C= \ln |\tan\frac{x}{2}| + C ∫cscxdx=∫sinx1dx=ln∣cscx−cotx∣+C=ln∣tan2x∣+C

常见凑微分：

∫ f ( a x + b ) d x = 1 a ∫ f ( a x + b ) d ( a x + b ) \int f(ax + b) \, dx = \frac{1}{a} \int f(ax + b) \, d(ax + b) ∫f(ax+b)dx=a1∫f(ax+b)d(ax+b) (其中 a ≠ 0 a \neq 0 a=0)
∫ f ( a x n + b ) x n − 1 d x = 1 n a ∫ f ( a x n + b ) d ( a x n + b ) \int f(ax^n + b) x^{n-1} \, dx = \frac{1}{na} \int f(ax^n + b) \, d(ax^n + b) ∫f(axn+b)xn−1dx=na1∫f(axn+b)d(axn+b) (其中 a ≠ 0 a \neq 0 a=0)
∫ f ( e x ) e x d x = ∫ f ( e x ) d e x \int f(e^x) e^x \, dx = \int f(e^x) \, de^x ∫f(ex)exdx=∫f(ex)dex
∫ f ( 1 x ) x 2 d x = − ∫ f ( 1 x ) d ( 1 x ) \int \frac{f\left(\frac{1}{x}\right)}{x^2} \, dx = - \int f\left(\frac{1}{x}\right) \, d\left(\frac{1}{x}\right) ∫x2f(x1)dx=−∫f(x1)d(x1)
∫ f ( ln ⁡ x ) x d x = ∫ f ( ln ⁡ x ) d ( ln ⁡ x ) \int \frac{f(\ln x)}{x} \, dx = \int f(\ln x) \, d(\ln x) ∫xf(lnx)dx=∫f(lnx)d(lnx)
∫ f ( x ) x d x = 2 ∫ f ( x ) d ( x ) \int \frac{f(\sqrt{x}) }{\sqrt{x}}\, dx = 2 \int f(\sqrt{x}) \, d(\sqrt{x}) ∫x f(x )dx=2∫f(x )d(x )
∫ f ( sin ⁡ x ) cos ⁡ x d x = ∫ f ( sin ⁡ x ) d ( sin ⁡ x ) \int f(\sin x) \cos x \, dx = \int f(\sin x) \, d(\sin x) ∫f(sinx)cosxdx=∫f(sinx)d(sinx)
∫ f ( cos ⁡ x ) sin ⁡ x d x = − ∫ f ( cos ⁡ x ) d ( cos ⁡ x ) \int f(\cos x) \sin x \, dx = -\int f(\cos x) \, d(\cos x) ∫f(cosx)sinxdx=−∫f(cosx)d(cosx)
∫ f ( tan ⁡ x ) sec ⁡ 2 x d x = ∫ f ( tan ⁡ x ) d ( tan ⁡ x ) \int f(\tan x) \sec^2 x \, dx = \int f(\tan x) \, d(\tan x) ∫f(tanx)sec2xdx=∫f(tanx)d(tanx)
∫ f ( cot ⁡ x ) csc ⁡ 2 x d x = − ∫ f ( cot ⁡ x ) d ( cot ⁡ x ) \int f(\cot x) \csc^2 x \, dx = -\int f(\cot x) \, d(\cot x) ∫f(cotx)csc2xdx=−∫f(cotx)d(cotx)
∫ f ( arcsin ⁡ x ) 1 − x 2 d x = ∫ f ( arcsin ⁡ x ) d ( arcsin ⁡ x ) \int \frac{f(\arcsin \sqrt{x}) }{\sqrt{1-x^2}} \, dx = \int f(\arcsin x) \, d(\arcsin x) ∫1−x2 f(arcsinx )dx=∫f(arcsinx)d(arcsinx)
∫ f ( arctan ⁡ x ) 1 + x 2 d x = ∫ f ( arctan ⁡ x ) d ( arctan ⁡ x ) \int \frac{f(\arctan x)}{1+x^2} \, dx = \int f(\arctan x) \, d(\arctan x) ∫1+x2f(arctanx)dx=∫f(arctanx)d(arctanx)

常见换元：

∫ a 2 − x 2 d x \int \sqrt{a^2 - x^2} \, dx ∫a2−x2 dx，令 x = a sin ⁡ t x = a \sin t x=asint，则 d x = a cos ⁡ t d t dx = a \cos t \, dt dx=acostdt
∫ a 2 + x 2 d x \int \sqrt{a^2 + x^2} \, dx ∫a2+x2 dx，令 x = a tan ⁡ t x = a \tan t x=atant，则 d x = a sec ⁡ 2 t d t dx = a \sec^2 t \, dt dx=asec2tdt
∫ x 2 − a 2 d x \int \sqrt{x^2 - a^2} \, dx ∫x2−a2 dx，令 x = a sec ⁡ t x = a \sec t x=asect，则 d x = a sec ⁡ t tan ⁡ t d t dx = a \sec t \tan t \, dt dx=asecttantdt

等价无穷小：

sin ⁡ x ∼ x \sin x \sim x sinx∼x
arcsin ⁡ x ∼ x \arcsin x \sim x arcsinx∼x
tan ⁡ x ∼ x \tan x \sim x tanx∼x
arctan ⁡ x ∼ x \arctan x \sim x arctanx∼x
ln ⁡ ( 1 + x ) ∼ x \ln(1 + x) \sim x ln(1+x)∼x
e x − 1 ∼ x e^x - 1 \sim x ex−1∼x
其他相似的等价无穷小关系

常见极限：

lim ⁡ x → 0 x α ln ⁡ β x = 0 \lim\limits_{x \to 0} x^\alpha \ln^\beta x = 0 x→0limxαlnβx=0，其中 α > 0 , β \alpha > 0, \beta α>0,β 为任意常数
lim ⁡ x → + ∞ x a e β x = 0 \lim\limits_{x \to +\infty} \frac{x^a} {e^{\beta x} }= 0 x→+∞limeβxxa=0，其中 α \alpha α 为任意常数， β > 0 \beta > 0 β>0
lim ⁡ x → + ∞ ln ⁡ β x x α = 0 \lim\limits_{x \to +\infty} \frac{\ln^\beta x}{x^\alpha} = 0 x→+∞limxαlnβx=0，其中 α > 0 , β \alpha > 0, \beta α>0,β 为任意常数
lim ⁡ n → ∞ n n = 1 \lim\limits_{n \to \infty} \sqrt[n]{n} = 1 n→∞limnn =1
lim ⁡ n → ∞ a n = 1 \lim\limits_{n \to \infty} \sqrt[n]{a} = 1 n→∞limna =1 (a>0)
lim ⁡ x → 0 + x x = 1 \lim\limits_{x \to 0^+} x^x = 1 x→0+limxx=1

泰勒公式：

f ( x ) = f ( x 0 ) + f ′ ( x 0 ) ( x − x 0 ) + 1 2 ! f ′ ′ ( x 0 ) ( x − x 0 ) 2 + ... + f ( n ) ( x 0 ) n ! ( x − x 0 ) n + R n ( x ) f(x) = f(x_0) + f'(x_0)(x - x_0) + \frac{1}{2!}f''(x_0)(x - x_0)^2 + \ldots + \frac{f^{(n)}(x_0)}{n!}(x - x_0)^n + R_n(x) f(x)=f(x0)+f′(x0)(x−x0)+2!1f′′(x0)(x−x0)2+...+n!f(n)(x0)(x−x0)n+Rn(x)

其中
R n ( x ) = f ( n + 1 ) ( ξ ) ( n + 1 ) ! ( x − x 0 ) n + 1 R_n(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!}(x - x_0)^{n+1} Rn(x)=(n+1)!f(n+1)(ξ)(x−x0)n+1

麦克劳林展开：

e x = 1 + x + 1 2 ! x 2 + ... + 1 n ! x n + o ( x n ) e^x = 1 + x + \frac{1}{2!}x^2 + \ldots + \frac{1}{n!}x^n + o(x^n) ex=1+x+2!1x2+...+n!1xn+o(xn)

ln ⁡ ( 1 + x ) = x − 1 2 x 2 + 1 3 x 3 − ... + ( − 1 ) n − 1 x n n + o ( x n ) \ln(1 + x) = x - \frac{1}{2}x^2 + \frac{1}{3}x^3 - \ldots + (-1)^{n-1}\frac{x^n}{n} + o(x^n) ln(1+x)=x−21x2+31x3−...+(−1)n−1nxn+o(xn)

sin ⁡ x = x − 1 3 ! x 3 + ... + ( − 1 ) n x 2 n + 1 ( 2 n + 1 ) ! + o ( x 2 n + 1 ) \sin x = x - \frac{1}{3!}x^3 + \ldots + (-1)^n \frac{x^{2n+1}}{(2n+1)!} + o(x^{2n+1}) sinx=x−3!1x3+...+(−1)n(2n+1)!x2n+1+o(x2n+1)

cos ⁡ x = 1 − 1 2 ! x 2 + ... + ( − 1 ) n x 2 n ( 2 n ) ! + o ( x 2 n ) \cos x = 1 - \frac{1}{2!}x^2 + \ldots + (-1)^n \frac{x^{2n}}{(2n)!} + o(x^{2n}) cosx=1−2!1x2+...+(−1)n(2n)!x2n+o(x2n)

tan ⁡ x = x + 1 3 x 3 + o ( x 3 ) \tan x = x + \frac{1}{3}x^3 + o(x^3) tanx=x+31x3+o(x3)

arcsin ⁡ x = x + 1 6 x 3 + o ( x 3 ) \arcsin x = x + \frac{1}{6}x^3 + o(x^3) arcsinx=x+61x3+o(x3)

1 1 − x = 1 + x + x 2 + ... + x n + o ( x n ) \frac{1}{1-x} = 1 + x + x^2 + \ldots + x^n + o(x^n) 1−x1=1+x+x2+...+xn+o(xn)

1 1 + x = 1 − x + x 2 + ... + ( − 1 ) n x n + o ( x n ) \frac{1}{1+x} = 1 - x + x^2 + \ldots + (-1)^n x^n + o(x^n) 1+x1=1−x+x2+...+(−1)nxn+o(xn)

( 1 + x ) m = 1 + m x + m ( m − 1 ) 2 ! x 2 + ... + m ( m − 1 ) ⋯ ( m − n + 1 ) n ! x n + o ( x n ) (1 + x)^m = 1 + mx + \frac{m(m-1)}{2!}x^2 + \ldots + \frac{m(m-1)\cdots(m-n+1)}{n!}x^n + o(x^n) (1+x)m=1+mx+2!m(m−1)x2+...+n!m(m−1)⋯(m−n+1)xn+o(xn)

间断点：

可去间断点: lim ⁡ x → x 0 f ( x ) \lim\limits_{x \to x_0} f(x) x→x0limf(x) 存在但不等于 f ( x 0 ) f(x_0) f(x0) 或 lim ⁡ x → x 0 f ( x ) \lim\limits_{x \to x_0} f(x) x→x0limf(x) 存在，但 f ( x ) f(x) f(x) 在 x 0 x_0 x0 处无定义。
跳跃间断点: lim ⁡ x → x 0 + f ( x ) \lim\limits_{x \to x_0^+} f(x) x→x0+limf(x), lim ⁡ x → x 0 − f ( x ) \lim\limits_{x \to x_0^-} f(x) x→x0−limf(x) 存在且不相等。
无穷间断点: lim ⁡ x → x 0 + f ( x ) = ∞ \lim\limits_{x \to x_0^+} f(x) = \infty x→x0+limf(x)=∞ 或 lim ⁡ x → x 0 − f ( x ) = ∞ \lim\limits_{x \to x_0^-} f(x) = \infty x→x0−limf(x)=∞。
震荡间断点: x → x 0 x \to x_0 x→x0, f ( x ) f(x) f(x) 在某区间内有无限多次变动。

渐近线：

水平渐近线: lim ⁡ x → + ∞ f ( x ) = a \lim\limits_{x \to +\infty} f(x) = a x→+∞limf(x)=a 或 lim ⁡ x → − ∞ f ( x ) = a \lim\limits_{x \to -\infty} f(x) = a x→−∞limf(x)=a。
铅直渐近线: 若 lim ⁡ x → x 0 − f ( x ) = ∞ \lim\limits_{x \to x_0^-} f(x) = \infty x→x0−limf(x)=∞ 或 lim ⁡ x → x 0 + f ( x ) = ∞ \lim\limits_{x \to x_0^+} f(x) = \infty x→x0+limf(x)=∞。
斜渐近线: k = lim ⁡ x → ∞ f ( x ) x k = \lim\limits_{x \to \infty} \frac{f(x)}{x} k=x→∞limxf(x), b = lim ⁡ x → ∞ [ f ( x ) − k x ] b = \lim\limits_{x \to \infty} [f(x) - kx] b=x→∞lim[f(x)−kx]。

微分中值定理：

费马定理：即极大极小值定义。
泰勒公式 ： f ( x ) = f ( x 0 ) + f ′ ( x 0 ) ( x − x 0 ) + 1 2 ! f ′ ′ ( x 0 ) ( x − x 0 ) 2 + ... + f ( n ) ( x 0 ) n ! ( x − x 0 ) n + f ( n + 1 ) ( ξ ) ( n + 1 ) ! ( x − x 0 ) n + 1 f(x) = f (x_0) + f'(x_0)(x - x_0) + \frac{1}{2!}f''(x_0)(x - x_0)^2 + \ldots + \frac{f^{(n)}(x_0)}{n!}(x - x_0)^n + \frac{f^{(n+1)}(\xi)}{(n+1)!}(x - x_0)^{n+1} f(x)=f(x0)+f′(x0)(x−x0)+2!1f′′(x0)(x−x0)2+...+n!f(n)(x0)(x−x0)n+(n+1)!f(n+1)(ξ)(x−x0)n+1。

在闭区间连续且开区间可导时：

罗尔定理 ： f ( a ) = f ( b ) f(a) = f(b) f(a)=f(b)。
拉格朗日中值定理 ： f ( b ) − f ( a ) b − a = f ′ ( ξ ) \frac{f(b)-f(a)}{b-a} = f'(\xi) b−af(b)−f(a)=f′(ξ)。
柯西中值定理 ： f ( b ) − f ( a ) g ( b ) − g ( a ) = f ′ ( ξ ) g ′ ( ξ ) \frac{f(b)-f(a)}{g(b)-g(a)} = \frac{f'(\xi)}{g'(\xi)} g(b)−g(a)f(b)−f(a)=g′(ξ)f′(ξ)，其中 g ′ ( x ) ≠ 0 g'(x) \neq 0 g′(x)=0。

定积分结论：

区间对称性 ： ∫ − l l f ( x ) d x = { 2 ∫ 0 1 f ( x ) d x , if f ( x ) is even 0 , if f ( x ) is odd \int_{-l}^{l} f(x)dx = \begin{cases} 2\int_0^1 f(x)dx, & \text{if } f(x) \text{ is even} \\ 0, & \text{if } f(x) \text{ is odd} \end{cases} ∫−llf(x)dx={2∫01f(x)dx,0,if f(x) is evenif f(x) is odd。
定积分换元 ： ∫ a b f ( x ) d x = ∫ a b f ( a + b − t ) d t = 1 2 ∫ a b [ f ( x ) + f ( a + b − x ) ] d x \int_{a}^{b} f(x)dx = \int_{a}^{b} f(a + b - t)dt = \frac{1}{2}\int_{a}^{b}[f(x) + f(a + b - x)]dx ∫abf(x)dx=∫abf(a+b−t)dt=21∫ab[f(x)+f(a+b−x)]dx。
周期函数 ： ∫ a + T a f ( x ) d x = ∫ T 0 f ( x ) d x = ∫ T 2 − T 2 f ( x ) d x \int_{a+T}^{a} f(x)dx = \int_{T}^{0} f(x)dx = \int_{\frac{T}{2}}^{-\frac{T}{2}} f(x)dx ∫a+Taf(x)dx=∫T0f(x)dx=∫2T−2Tf(x)dx。
华理士点火定理：
- ∫ 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 is even n − 1 n ⋅ n − 3 n − 2 ⋯ 2 3 ⋅ 1 , n > 1 is odd \int_0^{\frac{\pi}{2}} \sin^n xdx = \int_0^{\frac{\pi}{2}} \cos^n xdx = \begin{cases} \frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdots \frac{1}{2} \cdot \frac{\pi}{2}, & n \text{ is even} \\ \frac{n-1}{n} \cdot \frac{n-3}{n-2} \cdots \frac{2}{3} \cdot 1, & n > 1 \text{ is odd} \end{cases} ∫02πsinnxdx=∫02πcosnxdx={nn−1⋅n−2n−3⋯21⋅2π,nn−1⋅n−2n−3⋯32⋅1,n is evenn>1 is odd。
三角函数定积分：
- ∫ − π π sin ⁡ n x sin ⁡ m x d x = ∫ 0 2 π sin ⁡ n x sin ⁡ m x d x = { π , n = m 0 , n ≠ m \int_{-\pi}^{\pi} \sin nx \sin mx dx = \int_0^{2\pi} \sin nx \sin mx dx = \begin{cases} \pi, & n = m \\ 0, & n \neq m \end{cases} ∫−ππsinnxsinmxdx=∫02πsinnxsinmxdx={π,0,n=mn=m.
- ∫ − π π sin ⁡ n x cos ⁡ m x d x = ∫ 0 2 π sin ⁡ n x cos ⁡ m x d x = 0 \int_{-\pi}^{\pi} \sin nx \cos mx dx = \int_0^{2\pi} \sin nx \cos mx dx = 0 ∫−ππsinnxcosmxdx=∫02πsinnxcosmxdx=0.
- ∫ − π π cos ⁡ n x cos ⁡ m x d x = ∫ 0 2 π cos ⁡ n x cos ⁡ m x d x = { π , n = m 0 , n ≠ m \int_{-\pi}^{\pi} \cos nx \cos mx dx = \int_0^{2\pi} \cos nx \cos mx dx = \begin{cases} \pi, & n = m \\ 0, & n \neq m \end{cases} ∫−ππcosnxcosmxdx=∫02πcosnxcosmxdx={π,0,n=mn=m.

柯西不等式：

若 f ( x ) f(x) f(x) 和 g ( x ) g(x) g(x) 在闭区间 [ a , b ] [a, b] [a,b] 上连续且可积，则有
( ∫ a b f ( x ) g ( x ) d x ) 2 ≤ ( ∫ a b f 2 ( x ) d x ) ( ∫ a b g 2 ( x ) d x ) . \left( \int_a^b f(x)g(x)dx \right)^2 \leq \left( \int_a^b f^2(x)dx \right) \left( \int_a^b g^2(x)dx \right). (∫abf(x)g(x)dx)2≤(∫abf2(x)dx)(∫abg2(x)dx).

变限积分：

如果 F ( x ) = ∫ ϕ ( x ) a f ( t ) d t F(x) = \int_{\phi(x)}^{a} f(t)dt F(x)=∫ϕ(x)af(t)dt，则 F ′ ( x ) = − f [ ϕ ( x ) ] ⋅ ϕ ′ ( x ) F'(x) = -f[\phi(x)] \cdot \phi'(x) F′(x)=−f[ϕ(x)]⋅ϕ′(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 ) \frac{d}{dx} \left( \int_{\phi_1(x)}^{\phi_2(x)} f(t) dt \right) = f[\phi_2(x)] \phi_2'(x) - f[\phi_1(x)] \phi_1'(x) dxd(∫ϕ1(x)ϕ2(x)f(t)dt)=f[ϕ2(x)]ϕ2′(x)−f[ϕ1(x)]ϕ1′(x)。
F ( x ) = ∫ x a f ( x − t ) d t F(x) = \int_{x}^{a} f(x - t)dt F(x)=∫xaf(x−t)dt，设 u = x − t u = x - t u=x−t，那么 F ( x ) = ∫ x − a 0 f ( u ) d u F(x) = \int_{x-a}^{0} f(u)du F(x)=∫x−a0f(u)du，因此 F ′ ( x ) = − f ( x − a ) F'(x) = -f(x - a) F′(x)=−f(x−a).（设x>a,则x<t<a → -a<-t<-x → x-a<u<0）

反常积分：

∫ a + ∞ f ( x ) d x = lim ⁡ b → + ∞ ∫ a b f ( x ) d x \int_a^{+\infty} f(x)dx = \lim\limits_{b \to +\infty} \int_a^b f(x)dx ∫a+∞f(x)dx=b→+∞lim∫abf(x)dx
∫ − ∞ b f ( x ) d x = lim ⁡ a → − ∞ ∫ a b f ( x ) d x \int_{-\infty}^b f(x)dx = \lim\limits_{a \to -\infty} \int_a^b f(x)dx ∫−∞bf(x)dx=a→−∞lim∫abf(x)dx
∫ + ∞ − ∞ f ( x ) d x = ∫ c − ∞ f ( x ) d x + ∫ + ∞ c f ( x ) d x \int_{+\infty}^{-\infty} f(x)dx = \int_c^{-\infty} f(x)dx + \int_{+\infty}^c f(x)dx ∫+∞−∞f(x)dx=∫c−∞f(x)dx+∫+∞cf(x)dx
∫ a b f ( x ) d x = lim ⁡ ε → 0 + ∫ a + ε b f ( x ) d x = F ( b ) − lim ⁡ x → a + F ( a ) \int_a^b f(x)dx = \lim\limits_{ε \to 0^+} \int_{a+\varepsilon}^b f(x)dx=F(b)-\lim\limits_{x \to a^+}F(a) ∫abf(x)dx=ε→0+lim∫a+εbf(x)dx=F(b)−x→a+limF(a)
- 其中 x = a x = a x=a 是 f ( x ) f(x) f(x) 的瑕点, c=a+ε
∫ a b f ( x ) d x = lim ⁡ ε → 0 + ∫ a b − ε f ( x ) d x = lim ⁡ x → b − F ( x ) − F ( a ) \int_a^b f(x)dx = \lim\limits_{ε \to 0^+} \int_a^{b-\varepsilon} f(x)dx=\lim\limits_{x \to b^-}F(x)-F(a) ∫abf(x)dx=ε→0+lim∫ab−εf(x)dx=x→b−limF(x)−F(a)
- 其中 x = b x = b x=b 是 f ( x ) f(x) f(x) 的瑕点，c=b−ε
∫ a b f ( x ) d x = lim ⁡ ε → 0 + ∫ a c − ε f ( x ) d x + lim ⁡ η → 0 + ∫ c + η b f ( x ) d x = ∫ a c f ( x ) d x − ∫ c b f ( x ) d x \int_a^b f(x)dx = \lim\limits_{ε \to 0^+} \int_a^{c-\varepsilon} f(x)dx + \lim\limits_{\eta \to 0^+} \int_{c+\eta}^b f(x)dx=\int_a^c f(x)dx-\int_c^b f(x)dx ∫abf(x)dx=ε→0+lim∫ac−εf(x)dx+η→0+lim∫c+ηbf(x)dx=∫acf(x)dx−∫cbf(x)dx
- 其中 x = c x = c x=c 是 f ( x ) f(x) f(x) 的瑕点

特殊反常积分：

∫ 0 1 1 x p d x = { 收敛 , 0 < p < 1 发散 , p ≥ 1 \int_0^1 \frac{1}{x^p}dx= \begin{cases} 收敛, 0<p<1 \\ 发散, p \geq 1 \end{cases} ∫01xp1dx={收敛,0<p<1发散,p≥1
∫ 1 + ∞ 1 x p d x = { 收敛 , p > 1 发散 , p ≤ 1 \int_1^{+\infty} \frac{1}{x^p}dx= \begin{cases} 收敛, p > 1 \\ 发散, p \leq 1 \end{cases} ∫1+∞xp1dx={收敛,p>1发散,p≤1
∫ 2 + ∞ 1 x ln ⁡ p x d x = { 收敛 , p > 1 发散 , p ≤ 1 \int_2^{+\infty} \frac{1}{x \ln^p x}dx =\begin{cases} 收敛, p > 1 \\ 发散, p \leq 1 \end{cases} ∫2+∞xlnpx1dx={收敛,p>1发散,p≤1
∫ 1 + ∞ x k e − x p d x \int_1^{+\infty} {x^ke^{-x^p}}dx ∫1+∞xke−xpdx，其中 k k k 为常数且 p > 0 p > 0 p>0 时均收敛。
∫ 0 1 ln ⁡ k x x p d x \int_0^1 \frac{\ln^k x} { x^p} dx ∫01xplnkxdx，其中 k k k 为常数且 p < 1 p < 1 p<1 时均收敛。

面积/体积

两曲线之间的面积：
- S = ∫ a b ∣ f ( x ) − g ( x ) ∣ d x S = \int_{a}^{b} |f(x) - g(x)|dx S=∫ab∣f(x)−g(x)∣dx
旋转体体积：
- 绕 X 轴： V x = π ∫ a b f 2 ( x ) d x V_x = \pi \int_{a}^{b} f^2(x)dx Vx=π∫abf2(x)dx
- 绕 Y 轴： V y = 2 π ∫ a b ∣ x f ( x ) ∣ d x V_y = 2\pi \int_{a}^{b} |xf(x)|dx Vy=2π∫ab∣xf(x)∣dx
弧长/侧面积：
- 直角坐标系下： L : y = f ( x ) , a ≤ x ≤ b , l = ∫ a b 1 + y ′ 2 d x L:\ y = f(x),\ a \leq x \leq b,\ l = \int_{a}^{b} \sqrt{1 + y'^2}dx L: y=f(x), a≤x≤b, l=∫ab1+y′2 dx
- 参数方程下： L : { x = x ( t ) y = y ( t ) , a ≤ t ≤ b , l = ∫ a b [ x ′ ( t ) ] 2 + [ y ′ ( t ) ] 2 d t L:\begin{cases}\ x = x(t) \\y = y(t)\end{cases},\ a \leq t \leq b,\ l = \int_{a}^{b} \sqrt{[x'(t)]^2 + [y'(t)]^2}dt L:{ x=x(t)y=y(t), a≤t≤b, l=∫ab[x′(t)]2+[y′(t)]2 dt
- 极坐标系下： L : r = r ( θ ) , α ≤ θ ≤ β , l = ∫ α β r 2 + r ′ 2 d θ L:\ r = r(\theta),\ \alpha \leq \theta \leq \beta,\ l = \int_{\alpha}^{\beta} \sqrt{r^2 + r'^2}d\theta L: r=r(θ), α≤θ≤β, l=∫αβr2+r′2 dθ
计算平面曲线上以函数 f ( x ) f(x) f(x) 为高度的旋转曲线绕 X 轴旋转而成的体积。 s s s 表示平面曲线的弧长。
S = ∫ a b 2 π f ( x ) d s S = \int_{a}^{b} 2\pi f(x)ds S=∫ab2πf(x)ds