「考研数学」

考研数学基础核心计算

1、函数求极限

一、无穷小的比较

1、已知 lim ⁡ x → x 0 f ( x ) = 0 , lim ⁡ x → x 0 g ( x ) = 0 \lim_{x \to x0} f(x) = 0, \lim_{x \to x0}g(x) = 0 limxx0f(x)=0,limxx0g(x)=0

(1) lim ⁡ x → x 0 f ( x ) g ( x ) = 0 , \lim_{x \to x0} \frac{f(x)}{g(x)} = 0, limxx0g(x)f(x)=0,则称f(x)是g(x)在{x → x 0 \to x0 x0}时的高阶无穷小, 记作f(x) = o(g(x));

(2) lim ⁡ x → x 0 f ( x ) g ( x ) = ∞ \lim_{x \to x0} \frac{f(x)}{g(x)} = ∞ limxx0g(x)f(x)=,则称f(x)是g(x)在{x → x 0 \to x0 x0}时的低阶无穷小;

(3) lim ⁡ x → x 0 f ( x ) g ( x ) = k ≠ 0 \lim_{x \to x0} \frac{f(x)}{g(x)} = k \neq 0 limxx0g(x)f(x)=k=0,则称f(x)是g(x)在{x → x 0 \to x0 x0}时的同阶无穷小;\特别地, 若 lim ⁡ x → x 0 f ( x ) g ( x ) = 1 \lim _{x \to x0} \frac{f(x)}{g(x)} = 1 limxx0g(x)f(x)=1,则称f(x)是g(x)在{x → x 0 \to x0 x0}时的等阶无穷小。

二、常见等价无穷小(x → \to 0时)

  • x的1阶无穷小:
    (1) sin ⁡ x ∼ x \sin x \sim x sinxx
    (2) tan ⁡ x ∼ x \tan x \sim x tanxx
    (3 ) arcsin ⁡ x ∼ x )\arcsin x \sim x )arcsinxx
    (4) arctan ⁡ x ∼ x \arctan x \sim x arctanxx
    (5) ln ⁡ ( 1 + x ) ∼ x \ln(1 + x) \sim x ln(1+x)x
    (6) e x − 1 ∼ x e^{x} - 1 \sim x ex1x
    (7) a x − 1 ∼ x l n a a^{x} - 1 \sim xlna ax1xlna
    (8) 1 + x n − 1 ∼ 1 n x \sqrt[n]{1 + x} - 1 \sim \frac{1}{n}x n1+x 1n1x
    (9) ( 1 + x ) a − 1 ∼ a x (1 + x) ^ a - 1 \sim ax (1+x)a1ax

  • x的2阶无穷小:
    (1) 1 − cos ⁡ x ∼ 1 2 x 2 1 - \cos x \sim \frac{1}{2}x^2 1cosx21x2
    (2) 1 − cos ⁡ n x ∼ n 2 x 2 1 - \cos^nx \sim \frac{n}{2}x^2 1cosnx2nx2
    (3) ln ⁡ ( 1 + x ) − x ∼ 1 2 x 2 \ln(1 + x) - x \sim \frac{1}{2}x^2 ln(1+x)x21x2

  • x的3阶无穷小:
    (1) x − sin ⁡ x ∼ 1 6 x 3 x - \sin x \sim \frac{1}{6}x^3 xsinx61x3
    (2) tan ⁡ x − x ∼ 1 3 x 3 \tan x - x \sim \frac{1}{3}x^3 tanxx31x3
    (3) x − a r c s i n x ∼ 1 6 x 3 x - arcsinx \sim \frac{1}{6}x^3 xarcsinx61x3
    (4) arctan ⁡ x − x ∼ 1 3 x 3 \arctan x - x \sim \frac{1}{3}x^3 arctanxx31x3

三、等价替换原理

1、 若 α ∼ α ~ , β ∼ β ~ , 且 lim ⁡ β ~ α ~ 存在,则 lim ⁡ β α = lim ⁡ β ~ α ~ . 若\alpha \sim \tilde{\alpha} ,\beta \sim \tilde{\beta},且\lim{\frac{\tilde{\beta}}{\tilde{\alpha}}}存在,则\lim{\frac{\beta}{\alpha}} = \lim {\frac{\tilde{\beta}}{\tilde{\alpha}} }. αα~,ββ~,limα~β~存在,则limαβ=limα~β~.

证: lim ⁡ β α = lim ⁡ β β ~ ⋅ β ~ α ~ ⋅ α ~ α = lim ⁡ β ~ α ~ . 证:\lim{\frac{\beta}{\alpha}} = \lim{\frac{\beta}{\tilde{\beta}}\cdot\frac{\tilde{\beta}}{\tilde{\alpha}}\cdot\frac{\tilde{\alpha}}{\alpha}} = \lim{\frac{\tilde{\beta}}{\tilde{\alpha}}}. 证:limαβ=limβ~βα~β~αα~=limα~β~.

注意:等价定理说明等价无穷小只能用在相对于整个极限而言的乘除因子中,不可用在加减法中。

四、等价无穷小的充要条件

α ∼ β 的充分必要条件是 β = α + o ( α ) \alpha \sim \beta 的充分必要条件是\beta = \alpha + o(\alpha) αβ的充分必要条件是β=α+o(α)

五、泰勒公式

  • 1、麦克劳林公式(泰勒公式的特殊情形)
    f ( x ) = f ( 0 ) + f ′ ( 0 ) x + f ′ ′ ( 0 ) 2 ! x 2 + f ′ ′ ( 0 ) 3 ! x 3 + . . . . . . + f ( n ) ( 0 ) n ! x n + o ( x n ) f(x) = f(0) + {f}'(0)x + \frac{{f}''(0)}{2!}x^2 + \frac{{f}''(0)}{3!}x^3 + ...... + \frac{{f}^{(n)}(0)}{n!}{x}^n + o(x^n) f(x)=f(0)+f(0)x+2!f′′(0)x2+3!f′′(0)x3+......+n!f(n)(0)xn+o(xn)
  • 2、九个常见的泰勒公式
    (1) f ( x ) = sin ⁡ x = x − 1 6 x 3 + o ( x 3 ) f(x) = \sin x = x - \frac{1}{6}x^3 + o(x^3) f(x)=sinx=x61x3+o(x3)
    (2) f ( x ) = cos ⁡ x = 1 − 1 2 x 2 + 1 24 x 4 + o ( x 4 ) f(x) = \cos x = 1 - \frac{1}{2}x^2 + \frac{1}{24}x^4 + o(x^4) f(x)=cosx=121x2+241x4+o(x4)
    (3) f ( x ) = tan ⁡ x = x + 1 3 x 3 + o ( x 3 ) f(x) = \tan x = x + \frac{1}{3}x^3+o(x^3) f(x)=tanx=x+31x3+o(x3)
    (4) f ( x ) = arcsin ⁡ x = x + 1 6 x 3 + o ( x 3 ) f(x) = \arcsin x = x + \frac{1}{6}x^3 + o(x^3) f(x)=arcsinx=x+61x3+o(x3)
    (5) f ( x ) = arctan ⁡ x = x − 1 3 x 3 + o ( x 3 ) f(x) = \arctan x = x - \frac{1}{3}x^3 + o(x^3) f(x)=arctanx=x31x3+o(x3)
    (6) f ( x ) = e x = 1 + x + 1 2 x 2 + 1 6 x 3 + o ( x 3 ) f(x) = e ^x = 1 + x + \frac{1}{2}x^2 + \frac{1}{6}x^3 + o(x^3) f(x)=ex=1+x+21x2+61x3+o(x3)
    (7) f ( x ) = ln ⁡ ( 1 + x ) = x − 1 2 x 2 + 1 3 x 3 + o ( x 3 ) f(x) = \ln (1 + x) = x - \frac{1}{2}x^2 + \frac{1}{3}x^3 + o(x^3) f(x)=ln(1+x)=x21x2+31x3+o(x3)
    (8) f ( x ) = 1 1 − x = 1 + x + x 2 + x 3 + o ( x 3 ) f(x) = \frac{1}{1 - x} = 1 + x + x ^2 + x^3 + o(x^3) f(x)=1x1=1+x+x2+x3+o(x3)
    (9) f ( x ) = ( 1 + x ) α = 1 + α x + α ( α − 1 ) 2 ! x 2 + α ( α − 1 ) ( α − 2 ) 3 ! x 3 + o ( x 3 ) f(x) = (1 + x)^\alpha = 1 + \alpha x +\frac{ \alpha (\alpha - 1)}{2!}x^2 + \frac{\alpha (\alpha - 1) (\alpha - 2)}{3!}x^3+o(x^3) f(x)=(1+x)α=1+αx+2!α(α1)x2+3!α(α1)(α2)x3+o(x3)

六、极限运算法则

  • 定理1:有限个无穷小的和也是无穷小

  • 定理2:有界函数与无穷小的乘积是无穷小

  • 推论1:常数与无穷小的乘积是无穷小

  • 推论2:有限个无穷小的乘积是无穷小

  • 定理3:如果 lim ⁡ f ( x ) = A , lim ⁡ g ( x ) = B \lim f(x) = A, \lim g(x) = B limf(x)=A,limg(x)=B,那么
    (1) lim ⁡ [ f ( x ) ± lim ⁡ g ( x ) ] = A ± B \lim[f(x) \pm \lim g(x)] = A \pm B lim[f(x)±limg(x)]=A±B

    (2) l i m [ f ( x ) ⋅ g ( x ) ] = lim ⁡ f ( x ) ⋅ lim ⁡ g ( x ) = A ⋅ B lim[f(x) \cdot g(x)] = \lim f(x) \cdot \lim g(x) = A \cdot B lim[f(x)g(x)]=limf(x)limg(x)=AB

    (3)若又有B ≠ 0 , 则 lim ⁡ f ( x ) g ( x ) = lim ⁡ f ( x ) lim ⁡ g ( x ) = A B \neq 0, 则\lim \frac{f(x)}{g(x)} = \frac{\lim f(x)}{\lim g(x)} = \frac{A}{B} =0,limg(x)f(x)=limg(x)limf(x)=BA

    (4)若又有A,B不全为0,则 lim ⁡ f ( x ) g ( x ) = A B \lim f(x)^{g(x)} = A^{B} limf(x)g(x)=AB

  • 推论3:如果 lim ⁡ f ( x ) 存在 , 而 c 为常数 , 则 lim ⁡ [ c f ( x ) ] = c lim ⁡ f ( x ) . \lim f(x)存在,而c为常数,则\lim [cf(x)] = c \lim f(x). limf(x)存在,c为常数,lim[cf(x)]=climf(x).

  • 推论4:如果 lim ⁡ f ( x ) 存在 , 而 n 为正整数 , 则 lim ⁡ [ f ( x ) ] n = [ lim ⁡ f ( x ) ] n . \lim f(x)存在,而n为正整数,则\lim {[f(x)]}^n = {[\lim f(x)]}^n. limf(x)存在,n为正整数,lim[f(x)]n=[limf(x)]n.

  • 推论5(抓大头):
    ( 1 ) lim ⁡ P m ( x ) Q n ( x ) = lim ⁡ x → ∞ a 0 + a 1 x + a 2 x 2 + . . . + a m x m b 0 + b 1 x + b 2 x 2 + . . . + b n x n = { 0 , m < n a m b n , m = n ∞ , m > n . (1)\lim \frac{P_m(x)}{Q_n(x)} = \lim_{x\to\infty}\frac{a_0 + a_1x + a_2x^2 + ... + a_m x^m}{b_0+b_1x+b_2x^2+...+b_nx^n} = \left\{\begin{matrix} 0, m \lt n \\ \frac{a_m}{b_n} , m = n\\ \infty, m \gt n. \end{matrix}\right. (1)limQn(x)Pm(x)=xlimb0+b1x+b2x2+...+bnxna0+a1x+a2x2+...+amxm= 0,m<nbnam,m=n,m>n.
    ( 2 ) lim ⁡ x → 0 α m ( x ) β n ( x ) = lim ⁡ x → 0 α m x m + o ( x m ) b n x n + o ( x n ) = { ∞ , m < n , a m b n , m = n , a m , b n 均不为零 . 0 , m > n . (2)\lim_{x \to 0}\frac{\alpha_m(x)}{\beta_n(x)} = \lim_{x\to0}\frac{\alpha_mx^m + o(x^m)}{b_nx^n + o(x^n)} = \left\{\begin{matrix} \infty, m \lt n,\\ \frac{a_m}{b_n}, m = n, a_m,b_n均不为零.\\ 0, m \gt n. \end{matrix}\right. (2)x0limβn(x)αm(x)=x0limbnxn+o(xn)αmxm+o(xm)= ,m<n,bnam,m=n,am,bn均不为零.0,m>n.

  • 定理4:(复合函数的极限运算法则)设函数y = f[g(x)]是由函数u = g(x) 与函数y = f(u)复合而成,f(g(x))在点x_0的某去心邻域内有定义,若 lim ⁡ x → x 0 g ( x ) = u 0 , lim ⁡ u → u 0 f ( u ) = A \lim_{x\to{x_0}}g(x) = u_0,\lim_{u\to{u_0}}f(u) = A limxx0g(x)=u0,limuu0f(u)=A,且存在
    δ 0 > 0 \delta_0 \gt 0 δ0>0,当x ϵ U o ( x 0 , u 0 ) 时 , 有 g ( x ) ≠ u 0 \epsilon \stackrel{o}{U}(x_0, u_0)时,有g(x)\neq u_0 ϵUo(x0,u0),g(x)=u0,
    则 lim ⁡ x → x 0 f [ g ( x ) ] 则\lim_{x\to x_0}f[g(x)] limxx0f[g(x)] = lim ⁡ u → u 0 f ( u ) = A . \lim_{u\to u_0}f(u) = A. limuu0f(u)=A.

定理5:洛必达法则

(1) lim ⁡ x → x 0 f ( x ) g ( x ) 为 0 0 型或 ∞ ∞ 型 . \lim_{x \to x_0} \frac{f(x)}{g(x)}为\frac{0}{0}型或\frac{\infty}{\infty}型. limxx0g(x)f(x)00型或.
(2) 在 x = x 0 在x=x_0 x=x0的某去心邻域内,函数f(x),g(x)可导且 g ′ ( x ) ≠ 0. {g}'(x)\neq0. g(x)=0.
(3) lim ⁡ x → x 0 f ′ ( x ) g ′ ( x ) 存在或为无穷大 . \lim_{x\to x0}\frac{{f}'(x)}{{g}'(x)}存在或为无穷大. limxx0g(x)f(x)存在或为无穷大.

七、函数极限通法

求解极限的步骤:
  • (1)代入x的极限值,分析极限的类型和可使用的化简

  • (2)化简:

  • 1、根式有理化

  • 2、提(约)公因子

  • 3、计算非零因子

  • 4、等价无穷小替换

  • 5、拆分极限存在的项

  • 6、变量替换(尤其是倒代换)

  • 7、幂指函数指数化
    (3) 求值:

  • 1、洛必达法则

  • 2、泰勒公式

2、函数求导数

一、导数的定义

1、函数变化率

(1) f ′ ( x 0 ) = lim ⁡ Δ x → 0 f ( x 0 + Δ x ) − f ( x 0 ) Δ x {f}'(x_0) = \lim_{\Delta x \to 0} \frac{f(x_0 + \Delta x) - f(x_0)}{\Delta x} f(x0)=limΔx0Δxf(x0+Δx)f(x0)

(2) f ′ ( x 0 ) = lim ⁡ x → x 0 f ( x + x 0 ) − f ( x 0 ) x − x 0 {f}'(x_0) = \lim_{x \to x_0}\frac{f(x+x_0) - f(x_0)}{x - x_0} f(x0)=limxx0xx0f(x+x0)f(x0)

2、导数的几何意义
  • (1)切线的斜率

  • (2)切线方程:
    y − f ( x 0 ) = f ′ ( x 0 ) ( x − x 0 ) y - f(x_0) = {f}'(x_0)(x - x_0) yf(x0)=f(x0)(xx0)

  • (3)法线方程:
    y − f ( x 0 ) = − 1 f ′ ( x 0 ) ( x − x 0 ) y - f(x_0) = -\frac{1}{{f}'(x_0)}(x - x_0) yf(x0)=f(x0)1(xx0)

二、各类函数求导

  • 1、基本求导公式与四则运算
  • 2、复合函数求导
  • 3、隐函数求导
  • 4、参数方程求导
  • 5、反函数求导
  • 6、高阶导数

定积分

一、定积分的性质
1、线性性质

(1) ∫ a b [ f ( x ) + g ( x ) ] d x = ∫ a b f ( x ) d x = ∫ a b g ( x ) d x \int_{a}^{b}[f(x) + g(x)]\mathrm{d}x = \int_{a}^{b}f(x) \mathrm{d} x = \int_{a}^{b}g(x) \mathrm{d} x ab[f(x)+g(x)]dx=abf(x)dx=abg(x)dx

(2) ∫ a b f ( x ) d x = ∫ a c f ( x ) d x + ∫ c b f ( x ) d x \int_{a}^{b} f(x)\mathrm{d}x= \int_{a}^{c} f(x)\mathrm{d}x+ \int_{c}^{b}f(x)\mathrm{d}x abf(x)dx=acf(x)dx+cbf(x)dx

2、不等式性质

( 1 ) 若 f ( x ) ≤ g ( x ) 且 f ( x ) ≠ g ( x ) , 则 ∫ a b f ( x ) d x < ∫ a b g ( x ) d x (1)若f(x)\le g(x) 且f(x)\neq g(x),则\int_{a}^{b}f(x)\mathrm{d}x \lt \int_{a}^{b}g(x)\mathrm{d}x (1)f(x)g(x)f(x)=g(x),abf(x)dx<abg(x)dx
( 2 ) 若 m ≤ f ( x ) ≤ M , 则 m ( b − a ) ≤ ∫ a b f ( x ) d x ≤ M ( b − a ) . (2)若m \le f(x) \le M, 则m(b - a) \le \int_{a}^{b}f(x)\mathrm{d}x \le M(b - a). (2)mf(x)M,m(ba)abf(x)dxM(ba).
( 3 ) 积分中值定理 : ∫ a b f ( x ) d x = f ( ξ ) ⋅ ( b − a ) , ξ ε [ a , b ] (3)积分中值定理:\int_{a}^{b}f(x)\mathrm{d}x = f(\xi)\cdot(b - a) , \xi \varepsilon \left[ a, b\right ] (3)积分中值定理:abf(x)dx=f(ξ)(ba),ξε[a,b]
( 4 ) ∣ ∫ a b f ( x ) d x ∣ ≤ ∫ a b ∣ f ( x ) ∣ d x (4)\left | \int_{a}^{b} f(x)\mathrm{d}x\right| \le \int_{a}^{b} \left| f(x)\right|\mathrm{d} x (4) abf(x)dx abf(x)dx

3、对称性

( 1 ) 若 f ( x ) 为偶函数 , 则 ∫ − a a f ( x ) d x = 2 ⋅ ∫ 0 a f ( x ) d x (1) 若f(x)为偶函数,则\int_{-a}^{a}f(x)\mathrm{d}x = 2\cdot\int_{0}^{a}f(x)\mathrm{d} x (1)f(x)为偶函数,aaf(x)dx=20af(x)dx

( 2 ) 若 f ( x ) 为奇函数 , 则 ∫ − a a f ( x ) d x = 0. (2) 若 f(x)为奇函数,则\int_{-a}^{a}f(x)\mathrm{d}x = 0. (2)f(x)为奇函数,aaf(x)dx=0.

二、定积分的计算

1、牛顿莱布尼茨公式

∫ a b f ( x ) d x = F ( x ) ∣ a b = F ( b ) − F ( a ) \int_{a}^{b}f(x)\mathrm{d}x = \left .F(x)\right|_{a}^{b} = F(b) - F(a) abf(x)dx=F(x)ab=F(b)F(a)

2、定积分的换元法
3、定积分的分部积分法
4、区间在现公式:

∫ a b f ( x ) d x x = a + b − t ‾ ‾ ∫ a b f ( a + b − t ) d t = ∫ a b f ( a + b − x ) d x = 1 2 ∫ a b [ f ( x ) + f ( a + b − x ] d x \int_{a}^{b}f(x)\mathrm{d}x \underline{\underline{x = a + b - t}} \int_{a}^{b}f(a +b - t)\mathrm{d}t = \int_{a}^{b}f(a + b - x) \mathrm{d}x = \frac{1}{2}\int_{a}^{b}\left[ f(x) + f(a + b - x \right]\mathrm{d}x abf(x)dxx=a+btabf(a+bt)dt=abf(a+bx)dx=21ab[f(x)+f(a+bx]dx

5、华里士公式

三、不定积分

一、不定积分的概念与基本积分公式
1、概念

∫ f ( x ) d x = F ( x ) + C \int f(x) \mathrm{d}x = F(x) + C f(x)dx=F(x)+C

2、基本积分公式

( 1 ) C ′ = 0 (1){C}' = 0 (1)C=0

( 2 ) ∫ 0 d x = C (2)\int 0 \mathrm{d}x = C (2)0dx=C

( 3 ) ( x α ) ′ = α x α − 1 (3){(x ^ \alpha)}' = \alpha x^{\alpha - 1} (3)(xα)=αxα1

( 4 ) ∫ x α − 1 d x = 1 α ⋅ x a + c (4)\int x^{\alpha - 1}\mathrm{d}x =\frac{1}{\alpha}\cdot x^a +c (4)xα1dx=α1xa+c

( 5 ) sin ⁡ ′ x = cos ⁡ x (5){\sin}'x = \cos x (5)sinx=cosx

( 6 ) ∫ cos ⁡ x d x = sin ⁡ x + c (6)\int \cos x \mathrm{d}x = \sin x + c (6)cosxdx=sinx+c

( 7 ) cos ⁡ ′ x = − sin ⁡ x (7){\cos }' x = -\sin x (7)cosx=sinx

( 8 ) ∫ sin ⁡ x d x = − cos ⁡ x + c (8)\int \sin x \mathrm{d}x = -\cos x + c (8)sinxdx=cosx+c

( 9 ) tan ⁡ ′ x = sec ⁡ 2 x (9){\tan}' x = {\sec} ^ 2 x (9)tanx=sec2x

( 10 ) ∫ sec ⁡ 2 x d x = tan ⁡ x + c (10)\int {\sec }^2 x \mathrm{d}x = \tan x + c (10)sec2xdx=tanx+c

( 11 ) ( cot ⁡ x ) ′ = − csc ⁡ 2 x (11){(\cot x)}' = -{\csc } ^ 2 x (11)(cotx)=csc2x

( 12 ) ∫ csc ⁡ 2 x d x = − cot ⁡ x + c (12)\int {\csc}^2 x \mathrm{d}x = -\cot x + c (12)csc2xdx=cotx+c

( 13 ) ( sec ⁡ x ) ′ = sec ⁡ x ⋅ tan ⁡ x (13){(\sec x)}' = \sec x \cdot \tan x (13)(secx)=secxtanx

( 14 ) ∫ sec ⁡ x ⋅ tan ⁡ x d x = sec ⁡ x + c (14)\int \sec x \cdot \tan x \mathrm{d}x = \sec x + c (14)secxtanxdx=secx+c

( 15 ) ( csc ⁡ x ) ′ = − csc ⁡ x ⋅ cot ⁡ x (15){(\csc x)}' = -\csc x \cdot \cot x (15)(cscx)=cscxcotx

( 16 ) ∫ csc ⁡ x ⋅ cot ⁡ x d x = − csc ⁡ x + c (16)\int \csc x \cdot \cot x \mathrm{d}x = -\csc x + c (16)cscxcotxdx=cscx+c

( 17 ) ln ⁡ ∣ x ∣ ′ = 1 x (17){\ln \left | x\right |}' = \frac{1}{x} (17)lnx=x1

( 18 ) ∫ 1 x = ln ⁡ ∣ x ∣ + c (18)\int \frac{1}{x} = \ln \left | x \right | + c (18)x1=lnx+c

( 19 ) ( a x ) ′ = a x ⋅ ln ⁡ a ( a > 0 , a ≠ 1 ) (19){(a ^x)}' = a ^ x \cdot \ln a(a \gt 0, a \neq 1) (19)(ax)=axlna(a>0,a=1)

( 20 ) ∫ a x d x = 1 ln ⁡ a a x + c ( a > 0 , a ≠ 1 ) (20)\int a ^x \mathrm {d}x = \frac{1}{\ln a} a^x + c (a \gt 0, a \neq 1) (20)axdx=lna1ax+c(a>0,a=1)

( 21 ) ( e x ) ′ = e x (21){(e ^ x)}' = e ^ x (21)(ex)=ex

( 22 ) ∫ e x d x = e x + c (22)\int e ^x \mathrm {d}x = e ^x + c (22)exdx=ex+c

( 23 ) ( arcsin ⁡ x ) ′ = 1 1 − x 2 (23){(\arcsin x)}' = \frac{1}{\sqrt{1 - x^2}} (23)(arcsinx)=1x2 1

( 24 ) ∫ 1 1 − x 2 d x = arcsin ⁡ x + c (24)\int\frac{1}{\sqrt{1 - x ^ 2}}\mathrm{d}x = \arcsin x + c (24)1x2 1dx=arcsinx+c

( 25 ) ( arctan ⁡ x ) ′ = 1 1 + x 2 (25){(\arctan x)}' = \frac{1}{1 + x ^ 2} (25)(arctanx)=1+x21

( 26 ) ∫ 1 1 + x 2 d x = arctan ⁡ x + c (26)\int \frac{1}{1 + x^2}\mathrm{d}x = \arctan x + c (26)1+x21dx=arctanx+c

( 27 ) ( ln ⁡ ∣ x + x 2 + a ∣ ) ′ = 1 x 2 + a (27){(\ln \left | x + \sqrt{x ^ 2 + a}\right | )}' = \frac{1}{\sqrt{x ^ 2 + a}} (27)(ln x+x2+a )=x2+a 1

( 28 ) ∫ 1 x 2 + a = ln ⁡ ∣ x + x 2 + a ∣ + c (28)\int\frac{1}{\sqrt{x ^ 2 + a}} = \ln \left| x + \sqrt{x ^ 2 + a}\right| + c (28)x2+a 1=ln x+x2+a +c

二、四大积分方法

  • 1、第一类换元法(凑微分)

  • 2、第二类换元法(去根号)

  • 3、分部积分法

  • 4、有理函数积分法

三、三角有理函数积分

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