高等数学 常用数学公式

∫ 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 ∫cos2xdx​dx=∫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 ∫sin2xdx​dx=∫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=∫cos2xsinx​dx=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=∫sin2xcosx​dx=−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​=∫02π​​sinnxdx=∫02π​​cosnxdx=nn−1​In−2​

∫ 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+x2dx​dx=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−x2 ​dx​dx=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−x2 ​dx=2x​a2−x2 ​+2a2​arcsinax​+C

∫ 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−x2dx​dx=2a1​lna−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−a2dx​dx=2a1​lnx+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±a2 ​dx​=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+a2 ​dx=2x​x2+a2 ​+2a2​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−a2 ​dx=2x​x2−a2 ​−2a2​ln(x+x2−a2 ​)+C

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=loga​x,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−x2 ​1​

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​

lim ⁡ x → 0 sin ⁡ x x = 1 \lim _{x \to 0} \frac{\sin x}{x} = 1 x→0lim​xsinx​=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

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∼21​x2

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​(−∞