高等数学 常用数学公式
文章目录
- 1 基本积分表
- 1.1 三角函数相关
- 1.2 反三角函数相关
- 1.3 杂项
- 2 求导公式
- 3 重要极限
- 3.1 两个重要极限
- 3.2 常用的等价无穷小
- 3.3 泰勒展开式(函数的幂级数展开式)
- 4 分部积分法
- 5 华里士公式（点火公式）
- 6 伽马函数
- 6.1 函数形式
- 6.2 函数性质
- 6.2.1 递推公式
- 6.2.2 贝塔函数
- 6.2.3 伽马分布
- 6.2.4 余元公式
- 6.2.5 凹函数
1 基本积分表
1.1 三角函数相关
∫ tan x d x = − ln cos x + C \int \tan x dx = - \ln \cos x + C ∫tanxdx=−lncosx+C
∫ cot x d x = ln sin x + C \int \cot x dx = \ln \sin x + C ∫cotxdx=lnsinx+C
∫ sec x d x = ln sec x + tan x + C \int \sec x dx = \ln \sec x + \tan x + C ∫secxdx=lnsecx+tanx+C
∫ csc x d x = − ln csc x − cot x + C \int \csc x dx = - \ln \csc x - \cot x + C ∫cscxdx=−lncscx−cotx+C
∫ d x cos 2 x d x = ∫ sec 2 x d x = tan x + C \int \frac{dx}{\cos ^ 2 x} dx = \int \sec ^ 2 x dx = \tan x + C ∫cos2xdxdx=∫sec2xdx=tanx+C
∫ d x sin 2 x d x = ∫ csc 2 x d x = − cot x + C \int \frac{dx}{\sin ^ 2 x} dx = \int \csc ^ 2 x dx = -\cot x + C ∫sin2xdxdx=∫csc2xdx=−cotx+C
∫ sec x tan x d x = ∫ sin x cos 2 x d x = ln sec x + C \int \sec x \tan x dx = \int \frac{\sin x}{\cos ^ 2 x} dx = \ln \sec x + C ∫secxtanxdx=∫cos2xsinxdx=lnsecx+C
∫ csc x cot x d x = ∫ cos x sin 2 x d x = − ln csc x + C \int \csc x \cot x dx = \int \frac{\cos x}{\sin ^ 2 x} dx = -\ln \csc x + C ∫cscxcotxdx=∫sin2xcosxdx=−lncscx+C
I n = ∫ 0 π 2 sin n x d x = ∫ 0 π 2 cos n x d x = n − 1 n I n − 2 I_n = \int _{0} ^ {\frac{\pi}{2}} \sin ^ n x dx = \int _{0} ^ {\frac{\pi}{2}} \cos ^ n x dx = \frac{n - 1}{n} I _{n-2} In=∫02πsinnxdx=∫02πcosnxdx=nn−1In−2
1.2 反三角函数相关
∫ d x a 2 + x 2 d x = 1 a a r c t a n x a + C \int \frac{dx}{a ^ 2 + x ^ 2} dx = \frac{1}{a}\ arctan {\frac{x}{a}} + C ∫a2+x2dxdx=a1 arctanax+C
∫ d x a 2 − x 2 d x = arctan x a + C \int \frac{dx}{\sqrt{a ^ 2 - x ^ 2}} dx = \arctan {\frac{x}{a}} + C ∫a2−x2dxdx=arctanax+C
∫ a 2 − x 2 d x = x 2 a 2 − x 2 + a 2 2 arcsin x a + C \int \sqrt{a ^ 2 - x ^ 2} dx = \frac{x}{2} \sqrt{a ^ 2 - x ^ 2} + \frac{a ^2}{2}\arcsin {\frac{x}{a}} + C ∫a2−x2dx=2xa2−x2+2a2arcsinax+C
1.3 杂项
∫ d x a 2 − x 2 d x = 1 2 a ln a + x a − x + C \int \frac{dx}{a ^ 2 - x ^ 2} dx= \frac{1}{2a} \ln \frac{a + x}{a - x} + C ∫a2−x2dxdx=2a1lna−xa+x+C
∫ d x x 2 − a 2 d x = 1 2 a ln x − a x + a + C \int \frac{dx}{x ^ 2 - a ^ 2} dx= \frac{1}{2a} \ln \frac{x - a}{x + a} + C ∫x2−a2dxdx=2a1lnx+ax−a+C
∫ a x d x = a x ln a + C \int a ^ x dx = \frac{a ^x}{\ln a} + C ∫axdx=lnaax+C
∫ d x x 2 ± a 2 = ln ( x + x 2 ± a 2 ) + C \int \frac{dx}{\sqrt {x ^ 2 \pm a ^2}} = \ln (x + \sqrt{x ^ 2 \pm a ^2}) + C ∫x2±a2dx=ln(x+x2±a2)+C
∫ x 2 + a 2 d x = x 2 x 2 + a 2 + a 2 2 ln ( x + x 2 + a 2 ) + C \int \sqrt{x ^ 2 + a ^ 2} dx = \frac{x}{2} \sqrt{x ^ 2 + a ^ 2} + \frac{a ^ 2}{2} \ln (x + \sqrt{x ^2 + a ^ 2}) + C ∫x2+a2dx=2xx2+a2+2a2ln(x+x2+a2)+C
∫ x 2 − a 2 d x = x 2 x 2 − a 2 − a 2 2 ln ( x + x 2 − a 2 ) + C \int \sqrt{x ^ 2 - a ^ 2} dx = \frac{x}{2} \sqrt{x ^ 2 - a ^ 2} - \frac{a ^ 2}{2} \ln (x + \sqrt{x ^2 - a ^ 2}) + C ∫x2−a2dx=2xx2−a2−2a2ln(x+x2−a2)+C
2 求导公式
y = C , y ′ = 0 y = C, y' = 0 y=C,y′=0
y = x n , y ′ = n x n − 1 y = x^n, y' = nx^{n - 1} y=xn,y′=nxn−1
y = s i n x , y ′ = c o s x y = sinx, y' = cosx y=sinx,y′=cosx
y = c o s x , y ′ = − s i n x y = cosx, y' = -sinx y=cosx,y′=−sinx
y = t a n x , y ′ = 1 c o s 2 x = s e c 2 x y = tanx, y' = \frac{1}{cos^2x} = sec^2x y=tanx,y′=cos2x1=sec2x
y = c o t x , y ′ = − 1 s i n 2 x = − c s c 2 x y = cotx, y' = - \frac{1}{sin^2x} = -csc^2x y=cotx,y′=−sin2x1=−csc2x
y = s e c x , y ′ = s e c x ⋅ t a n x y = secx, y' = secx \cdot tanx y=secx,y′=secx⋅tanx
y = c s c x , y ′ = − c s c x ⋅ c o t x y = cscx, y' = -cscx \cdot cotx y=cscx,y′=−cscx⋅cotx
y = l n ∣ x ∣ , y ′ = 1 x y = ln|x|, y' = \frac{1}{x} y=ln∣x∣,y′=x1
y = l o g a x , y ′ = 1 x l n a y = log_a x, y' = \frac{1}{xlna} y=logax,y′=xlna1
y = e x , y ′ = e x y = e^x, y' = e^x y=ex,y′=ex
y = a x , y ′ = a x l n a ( a > 0 , a ≠ 1 ) y = a^x, y' = a^xlna (a > 0, a \ne 1) y=ax,y′=axlna(a>0,a̸=1)
y = a r c s i n x , y ′ = 1 1 − x 2 y = arcsinx, y' = \frac{1}{\sqrt{1 - x^2}} y=arcsinx,y′=1−x21
y = a r c t a n x , y ′ = 1 1 + x 2 y = arctanx, y' = \frac{1}{1 + x^2} y=arctanx,y′=1+x21
y = a r c c o t x , y ′ = − 1 1 + x 2 y = arccotx, y' = -\frac{1}{1 + x^2} y=arccotx,y′=−1+x21
3 重要极限
3.1 两个重要极限
lim x → 0 sin x x = 1 \lim _{x \to 0} \frac{\sin x}{x} = 1 x→0limxsinx=1
lim x → 0 ( 1 + x ) 1 x = lim x → ∞ ( 1 + 1 x ) x = e ≈ 2.71828 \lim _{x \to 0} (1 + x)^\frac{1}{x} = \lim _{x \to \infty } (1 + \frac{1}{x})^x = e \approx 2.71828 x→0lim(1+x)x1=x→∞lim(1+x1)x=e≈2.71828
3.2 常用的等价无穷小
sin x ∼ x , tan x ∼ x , arcsin x ∼ x , arctan x ∼ x \sin x \sim x, \tan x \sim x, \arcsin x \sim x, \arctan x \sim x sinx∼x,tanx∼x,arcsinx∼x,arctanx∼x
e x − 1 ∼ x , ln ( 1 + x ) ∼ x , ( 1 + x ) α − 1 ∼ α x , 1 − cos x ∼ 1 2 x 2 e ^ x - 1 \sim x, \ln (1 + x) \sim x, (1 + x) ^ \alpha - 1 \sim ~ \alpha x, 1 - \cos x \sim \frac{1}{2} x ^ 2 ex−1∼x,ln(1+x)∼x,(1+x)α−1∼ αx,1−cosx∼21x2
3.3 泰勒展开式(函数的幂级数展开式)
当 x → 0 x \to 0 x→0 时（作为幂级数展开式的 x x x的取值范围）
e x = 1 + x + x 2 2 ! + x 3 3 ! + ⋯ = ∑ n = 0 ∞ x n n ! ( − ∞ < x < + ∞ ) e ^ x = 1 + x + \frac{x ^ 2}{2!} + \frac{x ^ 3}{3!} + \dots = \sum _{n = 0} ^ {\infty} \frac{x ^ n}{n!} (- \infty < x < + \infty) ex=1+x+2!x2+3!x3+⋯=n=0∑∞n!xn(−∞<x<+∞)
sin x = x − x 3 3 ! + x 5 5 ! − ⋯ = ∑ k = 0 ∞ ( − 1 ) k x 2 k + 1 ( 2 k + 1 ) ! ( − ∞ < x < + ∞ ) \sin x = x - \frac{x ^ 3}{3!} + \frac{x ^ 5}{5!} - \dots = \sum _{k = 0} ^ {\infty} (-1) ^ k \frac{x ^ {2k + 1}}{(2k + 1)!} (- \infty < x < + \infty) sinx=x−3!x3+5!x5−⋯=k=0∑∞(−1)k(2k+1)!x2k+1(−∞<x<+∞)
cos x = 1 − x 2 2 ! + x 4 4 ! − ⋯ = ∑ k = 0 ∞ ( − 1 ) k x 2 k ( 2 k ) ! ( − ∞ < x < + ∞ ) \cos x = 1 - \frac{x ^ 2}{2!} + \frac{x ^ 4}{4!} - \dots = \sum _{k = 0} ^ {\infty} (-1) ^ k \frac{x ^ {2k}}{(2k)!} (- \infty < x < + \infty) cosx=1−2!x2+4!x4−⋯=k=0∑∞(−1)k(2k)!x2k(−∞<x<+∞)
ln ( 1 + x ) = x − x 2 2 + x 3 3 − ⋯ = ∑ n = 0 ∞ ( − 1 ) n + 1 x n n ( − 1 < x ≤ + 1 ) \ln(1 + x) = x - \frac{x ^ 2}{2} + \frac{x ^3}{3} - \dots = \sum _{n = 0} ^ {\infty} (-1) ^ {n + 1} \frac{x ^ {n}}{n} (- 1 < x \le + 1) ln(1+x)=x−2x2+3x3−⋯=n=0∑∞(−1)n+1nxn(−1<x≤+1)
arctan x = x − x 3 3 + x 5 5 − ⋯ = ∑ k = 0 ∞ ( − 1 ) k x 2 k + 1 2 k + 1 ( − 1 ≤ x ≤ + 1 ) \arctan x = x - \frac{x ^ 3}{3} + \frac{x ^ 5}{5} - \dots = \sum _{k = 0} ^ {\infty} (-1) ^ k \frac{x ^ {2k + 1}}{2k + 1} (- 1 \le x \le + 1) arctanx=x−3x3+5x5−⋯=k=0∑∞(−1)k2k+1x2k+1(−1≤x≤+1)
( 1 + x ) α = 1 + α x + α ( α − 1 ) 2 ! x 2 ⋯ = ∑ k = 0 ∞ α ( α − 1 ) … [ α − ( n − 1 ) ] n ! x n ( − 1 < x < + 1 ) (1 + x) ^ \alpha = 1 + \alpha x + \frac{\alpha (\alpha - 1)}{2!}x^2\dots = \sum _{k = 0} ^ {\infty} \frac{\alpha (\alpha - 1) \dots [\alpha - (n - 1)]}{n!} x ^ n (- 1 < x < + 1) (1+x)α=1+αx+2!α(α−1)x2⋯=k=0∑∞n!α(α−1)…[α−(n−1)]xn(−1<x<+1)
4 分部积分法
设 u ( x ) , v ( x ) u(x), v(x) u(x),v(x)均有连续的导数，则
∫ u ( x ) d v ( x ) = u ( x ) v ( x ) − ∫ v ( x ) d u ( x ) \int u(x)dv(x) = u(x)v(x) - \int v(x)du(x) ∫u(x)dv(x)=u(x)v(x)−∫v(x)du(x)
5 华里士公式（点火公式）
I n = ∫ 0 π 2 sin n x d x = ∫ 0 π 2 cos n x d x I_n = \int _0 ^ {\frac{\pi}{2}} \sin ^ n x dx = \int _0 ^{\frac{\pi}{2}} \cos ^ n x dx In=∫02πsinnxdx=∫02πcosnxdx
= { n − 1 n n − 3 n − 2 … 3 4 1 2 π 2 , n = 2 k + 2 n − 1 n n − 3 n − 2 … 4 5 2 3 , n = 2 k + 3 1 , n = 1 ( k ≥ 0 ) =\left\{ \begin{aligned} & \frac{n - 1}{n} \frac{n - 3}{n - 2} \dots \frac{3}{4} \frac{1}{2} \frac{\pi}{2} , n = 2k + 2\\ & \frac{n - 1}{n} \frac{n - 3}{n - 2} \dots \frac{4}{5} \frac{2}{3} , n = 2k + 3\\ & 1, n = 1 \end{aligned}(k \ge 0) \right. =⎩⎪⎪⎪⎪⎨⎪⎪⎪⎪⎧nn−1n−2n−3…43212π,n=2k+2nn−1n−2n−3…5432,n=2k+31,n=1(k≥0)
6 伽马函数
6.1 函数形式
含参变量 s ( s > 0 ) s(s> 0) s(s>0)的反常积分
Γ ( s ) = ∫ 0 + ∞ x s − 1 e − x d x , x > 0 \Gamma (s) = \int _ {0} ^ {+ \infty } x ^ {s - 1} e ^ {-x} dx, x > 0 Γ(s)=∫0+∞xs−1e−xdx,x>0
6.2 函数性质
6.2.1 递推公式
Γ ( x + 1 ) = x Γ ( x ) \Gamma (x + 1) = x \Gamma (x) Γ(x+1)=xΓ(x)
于是很容易证明，伽马函数可以当成是阶乘在实数集上的延拓，对于正整数n，具有如下性质：
Γ ( n ) = ( n − 1 ) ! \Gamma (n) = (n - 1)! Γ(n)=(n−1)!
6.2.2 贝塔函数
B ( m , n ) = Γ ( m ) Γ ( n ) Γ ( m + n ) B(m, n) = \frac{\Gamma(m) \Gamma (n)}{\Gamma (m + n)} B(m,n)=Γ(m+n)Γ(m)Γ(n)
6.2.3 伽马分布
在概率的研究中有一个重要的分布叫做伽玛分布：
f ( X ) = X α − 1 λ α e − λ X Γ ( α ) , X > 0 f(X) = \frac{X^{\alpha - 1} \lambda^{\alpha} e^{-\lambda X}}{\Gamma (\alpha )}, X > 0 f(X)=Γ(α)Xα−1λαe−λX,X>0
6.2.4 余元公式
对 x ∈ ( 0 , 1 ) x \in (0, 1) x∈(0,1)有
Γ ( 1 − x ) Γ ( x ) = π s i n π x \Gamma (1 - x) \Gamma(x) = \frac{\pi}{sin \pi x} Γ(1−x)Γ(x)=sinπxπ
这个公式称为余元公式。
由此可以推出以下重要的概率公式：
Γ ( 1 2 ) = π \Gamma (\frac{1}{2}) = \sqrt{\pi} Γ(21)=π
6.2.5 凹函数
对于 x > 0 x > 0 x>0，伽马函数是严格凹函数。