s
i
n
x
∼
x
sinx\sim x
sinx∼x,
t
a
n
x
∼
x
tanx\sim x
tanx∼x,
a
r
c
s
i
n
x
∼
x
arcsinx\sim x
arcsinx∼x,
a
r
c
t
a
n
x
∼
x
arctanx\sim x
arctanx∼x,
e
x
−
1
∼
x
e^x -1\sim x
ex−1∼x,
I
n
(
1
+
x
)
∼
x
In(1+x)\sim x
In(1+x)∼x,
a
x
−
1
=
e
x
I
n
a
−
1
∼
x
I
n
a
a^x-1= e^{xIna}-1\sim xIna
ax−1=exIna−1∼xIna,
1
−
c
o
s
x
∼
1
2
x
2
1-cosx\sim \frac{1}{2}x^2
1−cosx∼21x2,
(
1
+
x
)
a
−
1
∼
a
x
(1+x)^a -1\sim ax
(1+x)a−1∼ax,
小
+
大
∼
大
小+大\sim 大
小+大∼大,
∫
0
x
f
(
t
)
d
t
∼
x
\int_{0}^{x}f(t)dt\sim x
∫0xf(t)dt∼x
2 常用公式
2.1 和式夹逼准则的两个思路
n
→
∞
:
n
⋅
u
m
i
n
≤
∑
i
=
1
n
u
i
≤
n
⋅
u
m
a
x
n
→
有
限
:
1
⋅
u
m
a
x
≤
∑
i
=
1
n
u
i
≤
n
⋅
u
m
a
x
n\to \infty : n\cdot u_{min}\leq \sum\limits_{i=1}^{n}u_i \leq n\cdot u_{max}\\ n\to 有限 : 1\cdot u_{max}\leq \sum\limits_{i=1}^{n}u_i \leq n\cdot u_{max}
n→∞:n⋅umin≤i=1∑nui≤n⋅umaxn→有限:1⋅umax≤i=1∑nui≤n⋅umax
2.2 小基础
(
a
+
b
)
3
=
a
3
+
3
a
2
b
+
3
a
b
2
+
b
3
(a+b)^3 = a^3+ 3a^2b + 3ab^2 + b^3
(a+b)3=a3+3a2b+3ab2+b3
a
3
−
b
3
=
(
a
−
b
)
(
a
2
+
a
b
+
b
3
)
a^3-b^3 = (a-b)(a^2 + ab + b^3)
a3−b3=(a−b)(a2+ab+b3)
(
a
+
b
)
n
=
∑
k
=
0
n
C
n
k
a
n
−
k
b
k
(a+b)^n = \sum\limits_{k=0}^{n}C_n^ka^{n-k}b^k
(a+b)n=k=0∑nCnkan−kbk
∑
k
=
1
n
k
2
=
n
(
n
+
1
)
(
2
n
+
1
)
6
\sum\limits_{k=1}^{n}k^2 = \frac{n(n+1)(2n+1)}{6}
k=1∑nk2=6n(n+1)(2n+1)
∑
n
=
1
∞
1
n
2
=
π
2
6
\sum\limits_{n=1}^{\infty}\frac{1}{n^2} = \frac{\pi ^2}{6}
n=1∑∞n21=6π2
2.3 二元一次方程解
X
1
,
2
=
−
b
±
b
2
−
4
a
c
2
a
,
X
1
+
X
2
=
−
b
a
,
X
1
X
2
=
a
c
,
顶
点
:
(
−
b
2
a
,
c
−
b
2
4
a
)
X_{1,2} = \frac{-b\pm \sqrt{b^2-4ac}}{2a}, X_1 + X_2 = -\frac{b}{a}, X_1X_2 = \frac{a}{c},\\顶点: (-\frac{b}{2a} , c-\frac{b^2}{4a})
X1,2=2a−b±b2−4ac,X1+X2=−ab,X1X2=ca,顶点:(−2ab,c−4ab2)
3
★
\bigstar
★常用展开公式
★
\bigstar
★
e
x
=
1
+
x
+
x
2
2
!
+
⋯
=
∑
n
=
0
∞
x
n
n
!
e^x = 1+x+\frac{x^2}{2!}+\cdots =\sum\limits_{n=0}^{\infty}\frac{x^n}{n!}
ex=1+x+2!x2+⋯=n=0∑∞n!xn
I
n
(
1
+
x
)
=
x
−
x
2
2
+
⋯
=
∑
n
=
0
∞
(
−
1
)
n
−
1
x
n
n
(
−
1
<
x
≤
1
)
In(1+x) = x-\frac{x^2}{2}+\cdots =\sum\limits_{n=0}^{\infty}(-1)^{n-1}\frac{x^n}{n}\quad (-1<x\leq 1)
In(1+x)=x−2x2+⋯=n=0∑∞(−1)n−1nxn(−1<x≤1)
I
n
(
1
−
x
)
=
−
∑
n
=
0
∞
x
n
n
(
−
1
<
x
≤
1
)
In(1-x) = -\sum\limits_{n=0}^{\infty}\frac{x^n}{n}\quad (-1<x\leq 1)
In(1−x)=−n=0∑∞nxn(−1<x≤1)
1
1
−
x
=
1
+
x
+
x
2
+
⋯
=
∑
n
=
0
∞
x
n
∣
x
∣
<
1
\frac{1}{1-x} = 1+x+x^2+\cdots =\sum\limits_{n=0}^{\infty}x^n\quad \mid x\mid <1
1−x1=1+x+x2+⋯=n=0∑∞xn∣x∣<1
s
i
n
x
=
x
−
x
3
3
!
+
⋯
=
∑
n
=
0
−
∞
(
−
1
)
n
x
2
n
+
1
(
2
n
+
1
)
!
sinx = x-\frac{x^3}{3!}+\cdots =\sum\limits_{n=0}^{-\infty}(-1)^{n}\frac{x^{2n+1}}{(2n+1)!}
sinx=x−3!x3+⋯=n=0∑−∞(−1)n(2n+1)!x2n+1
c
o
s
x
=
1
−
x
2
2
!
+
⋯
=
∑
n
=
0
−
∞
(
−
1
)
n
x
2
n
(
2
n
)
!
cosx = 1-\frac{x^2}{2!}+\cdots =\sum\limits_{n=0}^{-\infty}(-1)^{n}\frac{x^{2n}}{(2n)!}
cosx=1−2!x2+⋯=n=0∑−∞(−1)n(2n)!x2n
t
a
n
x
=
x
+
x
3
3
+
O
(
x
3
)
tanx = x+\frac{x^3}{3}+O(x^3)
tanx=x+3x3+O(x3)
a
r
c
s
i
n
x
=
x
+
x
3
6
+
O
(
x
3
)
arcsinx = x+\frac{x^3}{6}+O(x^3)
arcsinx=x+6x3+O(x3)
a
r
c
t
a
n
x
=
x
−
x
3
3
+
O
(
x
3
)
=
∑
n
=
0
∞
(
−
1
)
n
x
2
n
+
1
2
n
+
1
arctanx = x-\frac{x^3}{3}+O(x^3) = \sum\limits_{n=0}^{\infty}(-1)^{n}\frac{x^{2n+1}}{2n+1}
arctanx=x−3x3+O(x3)=n=0∑∞(−1)n2n+1x2n+1
e
x
−
e
−
x
2
=
∑
n
=
0
∞
x
2
n
+
1
(
2
n
+
1
)
!
\frac{e^x-e^{-x}}{2} = \sum\limits_{n=0}^{\infty}\frac{x^{2n+1}}{(2n+1)!}
2ex−e−x=n=0∑∞(2n+1)!x2n+1
e
x
+
e
−
x
2
=
∑
n
=
0
∞
x
2
n
(
2
n
)
!
\frac{e^x+e^{-x}}{2} = \sum\limits_{n=0}^{\infty}\frac{x^{2n}}{(2n)!}
2ex+e−x=n=0∑∞(2n)!x2n
(
1
+
x
)
a
=
1
+
a
x
+
a
(
a
−
1
)
2
x
2
+
O
(
x
2
)
(1+x)^a = 1+ax+\frac{a(a-1)}2{x^2} + O(x^2)
(1+x)a=1+ax+2a(a−1)x2+O(x2)
4 常用不等式
a
r
c
t
a
n
x
<
x
<
a
r
c
s
i
n
x
(
0
≤
x
≤
1
)
arctanx < x < arcsinx \quad (0\leq x\leq 1)
arctanx<x<arcsinx(0≤x≤1)
e
x
≥
x
+
1
(
∀
x
)
e^x \geq x+1\quad (∀x)
ex≥x+1(∀x)
x
−
1
≥
I
n
x
(
x
>
0
)
x-1 \geq Inx\quad (x>0)
x−1≥Inx(x>0)
x
>
s
i
n
x
(
x
>
0
)
x> sinx\quad (x>0)
x>sinx(x>0)
1
1
+
x
<
I
n
(
1
+
1
x
)
<
1
x
\frac{1}{1+x}<In(1+\frac{1}{x})<\frac{1}{x}
1+x1<In(1+x1)<x1
x
1
+
x
<
I
n
(
1
+
x
)
<
x
\frac{x}{1+x}<In(1+x)<x
1+xx<In(1+x)<x
a
b
≤
a
+
b
2
≤
a
2
+
b
2
2
(
a
,
b
>
0
)
\sqrt{ab}\leq \frac{a+b}{2}\leq \sqrt{\frac{a^2+b^2}{2}}\quad (a,b>0)
ab≤2a+b≤2a2+b2(a,b>0)
a
b
c
3
≤
a
+
b
+
c
3
(
a
,
b
,
c
>
0
)
\sqrt[3]{abc}\leq \frac{a+b+c}{3}\quad (a,b,c>0)
3abc≤3a+b+c(a,b,c>0)
∣
a
±
b
∣
≤
∣
a
∣
+
∣
b
∣
\mid a\pm b\mid \leq \mid a\mid + \mid b\mid
∣a±b∣≤∣a∣+∣b∣
∣
∣
a
∣
−
∣
b
∣
∣
≤
∣
a
−
b
∣
\mid \mid a\mid - \mid b\mid \mid \leq \mid a-b\mid
∣∣a∣−∣b∣∣≤∣a−b∣
∣
∫
a
b
f
(
x
)
d
x
∣
≤
∫
a
b
∣
f
(
x
)
∣
d
x
\mid \int_a^bf(x)dx\mid \leq \int_a^b\mid f(x)\mid dx
∣∫abf(x)dx∣≤∫ab∣f(x)∣dx
5 三角变换
诱导公式法则：奇变偶不变，符号看象限！
s
i
n
2
x
=
2
s
i
n
x
c
o
s
x
sin2x = 2sinxcosx
sin2x=2sinxcosx
c
o
s
2
x
=
c
o
s
2
x
−
s
i
n
2
x
=
1
−
2
s
i
n
2
x
=
2
c
o
s
2
−
1
cos2x = cos^2x - sin^2x = 1-2sin^2x = 2cos^2-1
cos2x=cos2x−sin2x=1−2sin2x=2cos2−1
s
i
n
3
x
=
−
4
s
i
n
3
x
+
3
s
i
n
x
sin3x = -4sin^3x + 3sinx
sin3x=−4sin3x+3sinx
c
o
s
3
x
=
4
c
o
s
2
x
−
3
c
o
s
x
cos3x = 4cos^2x - 3cosx
cos3x=4cos2x−3cosx
s
i
n
x
⋅
c
o
s
y
=
1
2
[
s
i
n
(
x
+
y
)
+
s
i
n
(
x
−
y
)
]
sinx\cdot cosy = \frac{1}{2}[sin(x+y)+sin(x-y)]
sinx⋅cosy=21[sin(x+y)+sin(x−y)]
s
i
n
2
x
2
=
1
2
(
1
−
c
o
s
x
)
sin^2\frac{x}{2} = \frac{1}{2}(1-cosx)
sin22x=21(1−cosx)
c
o
s
2
x
2
=
1
2
(
1
+
c
o
s
x
)
cos^2\frac{x}{2} = \frac{1}{2}(1+cosx)
cos22x=21(1+cosx)
t
a
n
2
x
2
=
1
−
c
o
s
x
s
i
n
x
=
s
i
n
x
1
+
c
o
s
x
tan^2\frac{x}{2} = \frac{1-cosx}{sinx} = \frac{sinx}{1+cosx}
tan22x=sinx1−cosx=1+cosxsinx
s
i
n
x
=
2
t
a
n
x
2
1
+
t
a
n
2
x
2
sinx = \frac{2tan\frac{x}{2}}{1+tan^2\frac{x}{2}}
sinx=1+tan22x2tan2x
c
o
s
x
=
1
−
t
a
n
2
x
2
1
+
t
a
n
2
x
2
cosx = \frac{1-tan^2\frac{x}{2}}{1+tan^2\frac{x}{2}}
cosx=1+tan22x1−tan22x
t
a
n
2
x
=
2
t
a
n
x
1
−
t
a
n
2
x
tan2x = \frac{2tanx}{1-tan^2x}
tan2x=1−tan2x2tanx
c
o
t
2
x
=
c
o
t
2
x
−
1
2
c
o
t
x
cot2x = \frac{cot^2x - 1}{2cotx}
cot2x=2cotxcot2x−1
1
+
t
a
n
2
x
=
s
e
c
2
x
1+tan^2x = sec^2x
1+tan2x=sec2x
1
+
c
o
t
2
x
=
c
s
c
2
x
1+cot^2x = csc^2x
1+cot2x=csc2x
6
★
\bigstar
★微分
★
\bigstar
★
6.1 定义式
f
′
(
x
0
)
=
lim
Δ
x
→
0
f
(
x
0
+
Δ
x
)
−
f
(
x
0
)
Δ
x
=
lim
x
→
x
0
f
(
x
)
−
f
(
x
0
)
x
−
x
0
f'(x_0) = \lim\limits_{\Delta x\to{0}}\frac{f(x_0 + \Delta x) - f(x_0)}{\Delta x} = \lim\limits_{x\to{x_0}}\frac{f(x) - f(x_0)}{x-x_0}
f′(x0)=Δx→0limΔxf(x0+Δx)−f(x0)=x→x0limx−x0f(x)−f(x0)
f
(
n
)
(
x
0
)
=
lim
x
→
x
0
f
(
n
−
1
)
(
x
)
−
f
(
n
−
1
)
(
x
0
)
x
−
x
0
f^{(n)}(x_0) = \lim\limits_{x\to{x_0}} \frac{f^{(n-1)}(x) - f^{(n-1)}(x_0)}{x-x_0}
f(n)(x0)=x→x0limx−x0f(n−1)(x)−f(n−1)(x0)
(
t
a
n
x
)
′
=
s
e
c
2
x
(tanx)' = sec^2x
(tanx)′=sec2x
(
c
o
t
x
)
′
=
−
c
s
c
2
x
(cotx)' = -csc^2x
(cotx)′=−csc2x
(
s
e
c
x
)
′
=
s
e
c
x
t
a
n
x
,
(
c
s
c
x
)
′
=
−
c
s
c
x
c
o
t
x
(secx)' = secxtanx, (cscx)' = -cscxcotx
(secx)′=secxtanx,(cscx)′=−cscxcotx
(
a
r
c
s
i
n
x
)
′
=
1
1
−
x
2
,
(
a
r
c
c
o
s
x
)
′
=
−
1
1
−
x
2
(arcsinx)' = \frac{1}{\sqrt{1-x^2}},\quad (arccosx)' = -\frac{1}{\sqrt{1-x^2}}
(arcsinx)′=1−x21,(arccosx)′=−1−x21
(
a
r
c
t
a
n
x
)
′
=
1
1
+
x
2
,
(
a
r
c
c
o
t
x
)
′
=
−
1
1
+
x
2
(arctanx)' = \frac{1}{1+x^2},\quad (arccotx)' = -\frac{1}{1+x^2}
(arctanx)′=1+x21,(arccotx)′=−1+x21
(
I
n
∣
c
o
s
x
∣
)
′
=
−
t
a
n
x
(In\mid cosx\mid)' = -tanx
(In∣cosx∣)′=−tanx
(
I
n
∣
s
i
n
x
∣
)
′
=
c
o
t
x
(In\mid sinx\mid)' = cotx
(In∣sinx∣)′=cotx
(
I
n
∣
s
e
c
x
+
t
a
n
x
∣
)
′
=
s
e
c
x
(In\mid secx + tanx\mid)' = secx
(In∣secx+tanx∣)′=secx
(
I
n
∣
c
s
c
x
−
c
o
t
x
∣
)
′
=
c
s
c
x
(In\mid cscx - cotx\mid)' = cscx
(In∣cscx−cotx∣)′=cscx
[
I
n
(
x
+
x
2
±
a
2
)
]
′
=
1
x
2
±
a
2
[In(x+\sqrt{x^2\pm a^2})]' = \frac{1}{\sqrt{x^2\pm a^2}}
[In(x+x2±a2)]′=x2±a21
d
x
2
=
(
d
x
)
2
dx^2 = (dx)^2
dx2=(dx)2
d
(
x
2
)
=
2
x
d
x
d(x^2) = 2xdx
d(x2)=2xdx
(
u
v
w
)
′
=
u
′
v
w
+
u
v
′
w
+
u
v
w
′
(uvw)' = u'vw+uv'w+uvw'
(uvw)′=u′vw+uv′w+uvw′
(
u
v
)
(
n
)
=
∑
k
=
0
n
C
n
k
u
(
n
−
k
)
v
(
k
)
(uv)^{(n)} = \sum\limits_{k=0}^{n}C_n^ku^{(n-k)}v^{(k)}
(uv)(n)=k=0∑nCnku(n−k)v(k)
6.3 分析函数关注点
定义域、奇偶性、对称性、图形变换、单调性、极值、最值、凹凸性、拐点、三种渐近线（铅垂、水平、斜）
7
★
\bigstar
★积分
★
\bigstar
★
7.1 定积分定义
∫
a
b
f
(
x
)
d
x
=
lim
n
→
∞
∑
i
=
1
∞
f
(
a
+
b
−
a
n
i
)
b
−
a
n
\int_a^bf(x)dx =\lim\limits_{n\to{\infty}} \sum\limits_{i=1}^{\infty} f(a+\frac{b-a}{n}i)\frac{b-a}{n}
∫abf(x)dx=n→∞limi=1∑∞f(a+nb−ai)nb−a
∫
0
1
f
(
x
)
d
x
=
lim
n
→
∞
∑
i
=
1
∞
f
(
i
n
)
1
n
\int_0^1f(x)dx =\lim\limits_{n\to{\infty}} \sum\limits_{i=1}^{\infty} f(\frac{i}{n})\frac{1}{n}
∫01f(x)dx=n→∞limi=1∑∞f(ni)n1
∫
0
x
f
(
x
)
d
x
=
lim
n
→
∞
∑
i
=
1
∞
f
(
x
n
i
)
x
n
\int_0^xf(x)dx =\lim\limits_{n\to{\infty}} \sum\limits_{i=1}^{\infty} f(\frac{x}{n}i)\frac{x}{n}
∫0xf(x)dx=n→∞limi=1∑∞f(nxi)nx
7.2 基本积分表
∫
x
k
d
x
=
1
k
+
1
x
k
+
1
+
C
\int x^k dx = \frac{1}{k+1}x^{k+1} + C
∫xkdx=k+11xk+1+C
∫
a
x
d
x
=
a
x
I
n
a
+
C
\int a^x dx = \frac{a^x}{Ina} +C
∫axdx=Inaax+C
∫
s
i
n
x
d
x
=
−
c
o
s
+
C
\int sinx dx = -cos +C
∫sinxdx=−cos+C
∫
c
o
s
d
x
=
s
i
n
x
+
C
\int cos dx = sinx +C
∫cosdx=sinx+C
∫
t
a
n
x
d
x
=
−
I
n
∣
c
o
s
x
∣
+
C
\int tanx dx = -In\mid cosx\mid +C
∫tanxdx=−In∣cosx∣+C
∫
c
o
t
x
d
x
=
I
n
∣
s
i
n
x
∣
+
C
\int cotx dx = In\mid sinx\mid +C
∫cotxdx=In∣sinx∣+C
∫
s
e
c
x
d
x
=
I
n
∣
s
e
c
x
+
t
a
n
x
∣
+
C
\int secx dx = In\mid secx + tanx\mid +C
∫secxdx=In∣secx+tanx∣+C
∫
c
s
c
x
d
x
=
I
n
∣
c
s
c
x
−
c
o
t
x
∣
+
C
\int cscx dx = In\mid cscx - cotx\mid +C
∫cscxdx=In∣cscx−cotx∣+C
∫
s
e
c
2
x
d
x
=
t
a
n
x
+
C
\int sec^2x dx = tanx +C
∫sec2xdx=tanx+C
∫
c
s
c
2
x
d
x
=
−
c
o
t
x
+
C
\int csc^2x dx = -cotx +C
∫csc2xdx=−cotx+C
∫
s
e
c
x
t
a
n
x
d
x
=
s
e
c
x
+
C
\int secxtanx dx = secx +C
∫secxtanxdx=secx+C
∫
c
s
c
x
c
o
t
x
d
x
=
−
c
s
c
x
+
C
\int cscxcotx dx = -cscx +C
∫cscxcotxdx=−cscx+C
∫
1
1
−
x
2
d
x
=
a
r
c
s
i
n
x
+
C
\int \frac{1}{\sqrt{1-x^2}} dx = arcsinx +C
∫1−x21dx=arcsinx+C
∫
1
a
2
−
x
2
d
x
=
a
r
c
s
i
n
x
a
+
C
\int \frac{1}{\sqrt{a^2-x^2}} dx = arcsin\frac{x}{a} +C
∫a2−x21dx=arcsinax+C
∫
1
1
+
x
2
d
x
=
a
r
c
t
a
n
x
+
C
\int \frac{1}{1+x^2} dx = arctanx +C
∫1+x21dx=arctanx+C
∫
1
a
2
+
x
2
d
x
=
1
a
a
r
c
t
a
n
x
a
+
C
(
a
>
0
)
\int \frac{1}{a^2+x^2} dx = \frac{1}{a}arctan\frac{x}{a} +C\quad (a>0)
∫a2+x21dx=a1arctanax+C(a>0)
∫
1
x
2
+
a
2
d
x
=
I
n
(
x
+
x
2
+
a
2
)
+
C
\int \frac{1}{\sqrt{x^2+a^2}} dx = In(x+\sqrt{x^2+a^2}) +C
∫x2+a21dx=In(x+x2+a2)+C
∫
1
x
2
−
a
2
d
x
=
I
n
(
x
+
x
2
−
a
2
)
+
C
(
∣
x
∣
>
∣
a
∣
)
\int \frac{1}{\sqrt{x^2-a^2}} dx = In(x+\sqrt{x^2-a^2}) +C\quad (\mid x\mid>\mid a\mid)
∫x2−a21dx=In(x+x2−a2)+C(∣x∣>∣a∣)
∫
1
x
2
−
a
2
d
x
=
1
2
a
I
n
∣
x
−
a
x
+
a
∣
+
C
\int \frac{1}{x^2-a^2} dx = \frac{1}{2a}In\mid \frac{x-a}{x+a}\mid +C
∫x2−a21dx=2a1In∣x+ax−a∣+C
∫
1
a
2
−
x
2
d
x
=
1
2
a
I
n
∣
x
+
a
x
−
a
∣
+
C
\int \frac{1}{a^2-x^2} dx = \frac{1}{2a}In\mid \frac{x+a}{x-a}\mid +C
∫a2−x21dx=2a1In∣x−ax+a∣+C
∫
a
2
−
x
2
d
x
=
a
2
2
a
r
c
s
i
n
x
a
+
x
2
a
2
−
x
2
+
C
(
∣
x
∣
<
a
)
\int \sqrt{a^2-x^2} dx = \frac{a^2}{2}arcsin\frac{x}{a} + \frac{x}{2}\sqrt{a^2-x^2} +C\quad (\mid x\mid<a)
∫a2−x2dx=2a2arcsinax+2xa2−x2+C(∣x∣<a)
∫
s
i
n
2
x
d
x
=
x
2
−
s
i
n
2
x
4
+
C
\int sin^2x dx = \frac{x}{2} -\frac{sin2x}{4} +C
∫sin2xdx=2x−4sin2x+C
∫
c
o
s
2
x
d
x
=
x
2
+
s
i
n
2
x
4
+
C
\int cos^2x dx = \frac{x}{2} +\frac{sin2x}{4} +C
∫cos2xdx=2x+4sin2x+C
7.3 常用积分公式
∫
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
∫abf(x)dx=∫abf(a+b−x)dx
∫
a
b
f
(
x
)
d
x
=
1
2
∫
a
b
[
f
(
x
)
+
f
(
a
+
b
−
x
)
]
d
x
\int_a^b f(x) dx = \frac{1}{2} \int_a^b [f(x)+f(a+b-x)] dx
∫abf(x)dx=21∫ab[f(x)+f(a+b−x)]dx
∫
a
b
f
(
x
)
d
x
=
∫
a
a
+
b
2
[
f
(
x
)
+
f
(
a
+
b
−
x
)
]
d
x
\int_a^b f(x) dx = \int_a^{\frac{a+b}{2}} [f(x)+f(a+b-x)] dx
∫abf(x)dx=∫a2a+b[f(x)+f(a+b−x)]dx
点火公式：
∫
0
π
2
s
i
n
n
x
d
x
=
∫
0
π
2
c
o
s
n
x
d
x
=
n
−
1
n
n
−
3
n
−
2
⋯
\int_0^\frac{\pi}{2}sin^nxdx = \int_0^\frac{\pi}{2}cos^nxdx = \frac{n-1}{n} \frac{n-3}{n-2} \cdots
∫02πsinnxdx=∫02πcosnxdx=nn−1n−2n−3⋯
∫
0
π
x
f
(
s
i
n
x
)
d
x
=
π
2
∫
0
π
f
(
s
i
n
x
)
d
x
=
π
∫
0
π
2
f
(
s
i
n
x
)
\int_0^{\pi} xf(sinx) dx = \frac{\pi}{2}\int_0^{\pi} f(sinx) dx =\pi \int_0^{\frac{\pi}{2}} f(sinx)
∫0πxf(sinx)dx=2π∫0πf(sinx)dx=π∫02πf(sinx)
∫
0
π
2
f
(
s
i
n
x
)
d
x
=
∫
0
π
2
f
(
c
o
s
x
)
d
x
\int_0^{\frac{\pi}{2}} f(sinx) dx = \int_0^{\frac{\pi}{2}} f(cosx) dx
∫02πf(sinx)dx=∫02πf(cosx)dx
∫
0
π
2
f
(
s
i
n
x
,
c
o
s
x
)
d
x
=
∫
0
π
2
f
(
c
o
s
x
,
s
i
n
x
)
d
x
\int_0^{\frac{\pi}{2}} f(sinx, cosx) dx = \int_0^{\frac{\pi}{2}} f(cosx,sinx) dx
∫02πf(sinx,cosx)dx=∫02πf(cosx,sinx)dx
∫
a
b
f
(
x
)
d
x
=
∫
−
π
2
π
2
f
(
a
+
b
2
+
b
−
a
2
s
i
n
t
)
⋅
b
−
a
2
c
o
s
t
d
t
,
(
x
−
a
+
b
2
=
b
−
a
2
s
i
n
t
)
\int_a^b f(x) dx = \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} f(\frac{a+b}{2} + \frac{b-a}{2} sint)\cdot \frac{b-a}{2}cost dt, \quad (x-\frac{a+b}{2} = \frac{b-a}{2}sint)
∫abf(x)dx=∫−2π2πf(2a+b+2b−asint)⋅2b−acostdt,(x−2a+b=2b−asint)
∫
a
b
f
(
x
)
d
x
=
∫
0
1
(
b
−
a
)
f
[
a
+
(
b
−
a
)
t
]
d
t
,
(
x
−
a
=
(
b
−
a
)
t
)
\int_a^b f(x) dx = \int_{0}^{1} (b-a) f[a+(b-a)t] dt, \quad (x-a = (b-a)t)
∫abf(x)dx=∫01(b−a)f[a+(b−a)t]dt,(x−a=(b−a)t)
∫
−
a
a
f
(
x
)
d
x
=
∫
0
a
[
f
(
x
)
+
f
(
−
x
)
]
d
x
\int_{-a}^a f(x) dx = \int_{0}^{a} [f(x) + f(-x)] dx
∫−aaf(x)dx=∫0a[f(x)+f(−x)]dx
∫
0
n
π
x
∣
s
i
n
x
∣
d
x
=
n
2
π
\int_0^{n\pi} x\mid sinx\mid dx = n^2\pi
∫0nπx∣sinx∣dx=n2π