Most commonly used high number formula

1 The equivalent infinitesimal ($x\to 0$)

$sinx\sim x$, $tanx\sim x$, $arcsinx\sim x$, $arctanx\sim x$, $e^x-1\sim x$, $In(1+x)\sim x$,
$a^x-1=e^{xIna}-1\sim xIna$, $1-cosx\sim \frac{1}{2}{x}^2$, $(1+x{)}^a-1\sim ax$,
$Small+Big\sim Big$, ${\int }_0^{x}f(t)dt\sim x$

2 Common formula

2.1 Two ideas of the sum pinch rule

$$\begin{gathered} n\to \infty :n\cdot u_{min}\le \sum _{i=1}^n{u}_i\le n\cdot u_{max} \\ n\to Yeslimit:1\cdot u_{max}\le \sum _{i=1}^n{u}_i\le n\cdot u_{max} \end{gathered}$$

2.2 Small foundation

1. $(a+b{)}^3=a^3+3a^{2}b+3a{b}^2+b^3$
2. $a^3-b^3=(a-b)(a^2+ab+b^3)$
3. $(a+b{)}^n=\sum _{k=0}^n{C}_n^k{a}^{n-k}{b}^k$
4. $\sum _{k=1}^n{k}^2=\frac{n(n+1)(2n+1)}{6}$
5. $\sum _{n=1}^{\infty }\frac{1}{n^2}=\frac{{\pi }^2}{6}$

2.3 Solution of quadratic equation of one variable

$$\begin{gathered} X_{1,2}=\frac{-b\pm \sqrt{b^2-4ac}}{2a},X_1+X_2=-\frac{b}{a},X_1{X}_2=\frac{a}{c}, \\ The\; topspot:(-\frac{b}{2a},c-\frac{b^2}{4a}) \end{gathered}$$

3 $*$ Common expansion formula $*$

1. $e^x=1+x+\frac{x^2}{2!}+\cdots =\sum _{n=0}^{\infty }\frac{x^n}{n!}$
2. $In(1+x)=x-\frac{x^2}{2}+\cdots =\sum _{n=0}^{\infty }(-1{)}^{n-1}\frac{x^n}{n}(-1<x\le 1)$
3. $In(1-x)=-\sum _{n=0}^{\infty }\frac{x^n}{n}(-1<x\le 1)$
4. $\frac{1}{1-x}=1+x+x^2+\cdots =\sum _{n=0}^{\infty }{x}^n\mid x\mid <1$
5. $sinx=x-\frac{x^3}{3!}+\cdots =\sum _{n=0}^{-\infty }(-1{)}^n\frac{x^{2n+1}}{(2n+1)!}$
6. $cosx=1-\frac{x^2}{2!}+\cdots =\sum _{n=0}^{-\infty }(-1{)}^n\frac{x^{2n}}{(2n)!}$
7. $tanx=x+\frac{x^3}{3}+O(x^3)$
8. $arcsinx=x+\frac{x^3}{6}+O(x^3)$
9. $arctanx=x-\frac{x^3}{3}+O(x^3)=\sum _{n=0}^{\infty }(-1{)}^n\frac{x^{2n+1}}{2n+1}$
10. $\frac{e^x-e^{-x}}{2}=\sum _{n=0}^{\infty }\frac{x^{2n+1}}{(2n+1)!}$
11. $\frac{e^x+e^{-x}}{2}=\sum _{n=0}^{\infty }\frac{x^{2n}}{(2n)!}$
12. $(1+x{)}^a=1+ax+\frac{a(a-1)}{2}{x}^2+O(x^2)$

4 Common inequalities

1. $arctanx<x<arcsinx(0\le x\le 1)$
2. $e^x\ge x+1(\forall x)$
3. $x-1\ge Inx(x>0)$
4. $x>sinx(x>0)$
5. $\frac{1}{1+x}<In(1+\frac{1}{x})<\frac{1}{x}$
6. $\frac{x}{1+x}<In(1+x)<x$
7. $\sqrt{ab}\le \frac{a+b}{2}\le \sqrt{\frac{a^2+b^2}{2}}(a,b>0)$
8. $\sqrt[3]{abc}\le \frac{a+b+c}{3}(a,b,c>0)$
9. $\mid a\pm b\mid \le \mid a\mid +\mid b\mid$
10. $\mid \mid a\mid -\mid b\mid \mid \le \mid a-b\mid$
11. $\mid {\int }_a^{b}f(x)dx\mid \le {\int }_a^b\mid f(x)\mid dx$

5 Trigonometric transformation

1. Induction formula rule ： Odd change and even change , Look at the quadrant ！
2. $sin2x=2sinxcosx$
3. $cos2x=co{s}^{2}x-si{n}^{2}x=1-2si{n}^{2}x=2co{s}^2-1$
4. $sin3x=-4si{n}^{3}x+3sinx$
5. $cos3x=4co{s}^{2}x-3cosx$
6. $sinx\cdot cosy=\frac{1}{2}[sin(x+y)+sin(x-y)]$
7. $si{n}^2\frac{x}{2}=\frac{1}{2}(1-cosx)$
8. $co{s}^2\frac{x}{2}=\frac{1}{2}(1+cosx)$
9. $ta{n}^2\frac{x}{2}=\frac{1-cosx}{sinx}=\frac{sinx}{1+cosx}$
10. $sinx=\frac{2tan\frac{x}{2}}{1+ta{n}^2\frac{x}{2}}$
11. $cosx=\frac{1-ta{n}^2\frac{x}{2}}{1+ta{n}^2\frac{x}{2}}$
12. $tan2x=\frac{2tanx}{1-ta{n}^{2}x}$
13. $cot2x=\frac{co{t}^{2}x-1}{2cotx}$
14. $1+ta{n}^{2}x=se{c}^{2}x$
15. $1+co{t}^{2}x=cs{c}^{2}x$

6 $*$ differential $*$

6.1 Definition

• $$f^{\prime }(x_0)=\underset{\Delta x\to 0}{\lim }\frac{f(x_0+\Delta x)-f(x_0)}{\Delta x}=\underset{x\to x_0}{\lim }\frac{f(x)-f(x_0)}{x-x_0}$$
• $$f^{(n)}(x_0)=\underset{x\to x_0}{\lim }\frac{f^{(n-1)}(x)-f^{(n-1)}(x_0)}{x-x_0}$$
• continuity $\nRightarrow$ Derivable , Derivable $\Rightarrow$ continuity , Derivable $\iff$ It's very small

6.2 It is often difficult to remember differential formulas

1. $(tanx{)}^{\prime }=se{c}^{2}x$
2. $(cotx{)}^{\prime }=-cs{c}^{2}x$
3. $(secx{)}^{\prime }=secxtanx,(cscx{)}^{\prime }=-cscxcotx$
4. $(arcsinx{)}^{\prime }=\frac{1}{\sqrt{1-x^2}},(arccosx{)}^{\prime }=-\frac{1}{\sqrt{1-x^2}}$
5. $(arctanx{)}^{\prime }=\frac{1}{1+x^2},(arccotx{)}^{\prime }=-\frac{1}{1+x^2}$
6. $(In\mid cosx\mid {)}^{\prime }=-tanx$
7. $(In\mid sinx\mid {)}^{\prime }=cotx$
8. $(In\mid secx+tanx\mid {)}^{\prime }=secx$
9. $(In\mid cscx-cotx\mid {)}^{\prime }=cscx$
10. $[In(x+\sqrt{x^2\pm a^2}){]}^{\prime }=\frac{1}{\sqrt{x^2\pm a^2}}$
11. $d{x}^2=(dx{)}^2$
12. $d(x^2)=2xdx$
13. $(uvw{)}^{\prime }=u^{\prime }vw+u{v}^{\prime }w+uv{w}^{\prime }$
14. $(uv{)}^{(n)}=\sum _{k=0}^n{C}_n^k{u}^{(n-k)}{v}^{(k)}$

6.3 Analyze function concerns

Domain of definition 、 Parity 、 symmetry 、 Graphic transformation 、 monotonicity 、 extremum 、 The most value 、 Concavity and convexity 、 Inflection point 、 Three asymptotes （ Plumb 、 level 、 oblique ）

7 $*$ integral $*$

7.1 The definition of definite integral

• ${\int }_a^{b}f(x)dx=\underset{n\to \infty }{\lim }\sum _{i=1}^{\infty }f(a+\frac{b-a}{n}i)\frac{b-a}{n}$
• ${\int }_0^{1}f(x)dx=\underset{n\to \infty }{\lim }\sum _{i=1}^{\infty }f(\frac{i}{n})\frac{1}{n}$
• ${\int }_0^{x}f(x)dx=\underset{n\to \infty }{\lim }\sum _{i=1}^{\infty }f(\frac{x}{n}i)\frac{x}{n}$

7.2 The basic integral table

1. $\int x^{k}dx=\frac{1}{k+1}{x}^{k+1}+C$
2. $\int a^{x}dx=\frac{a^x}{Ina}+C$
3. $\int sinxdx=-cos+C$
4. $\int cosdx=sinx+C$
5. $\int tanxdx=-In\mid cosx\mid +C$
6. $\int cotxdx=In\mid sinx\mid +C$
7. $\int secxdx=In\mid secx+tanx\mid +C$
8. $\int cscxdx=In\mid cscx-cotx\mid +C$
9. $\int se{c}^{2}xdx=tanx+C$
10. $\int cs{c}^{2}xdx=-cotx+C$
11. $\int secxtanxdx=secx+C$
12. $\int cscxcotxdx=-cscx+C$
13. $\int \frac{1}{\sqrt{1-x^2}}dx=arcsinx+C$
14. $\int \frac{1}{\sqrt{a^2-x^2}}dx=arcsin\frac{x}{a}+C$
15. $\int \frac{1}{1+x^2}dx=arctanx+C$
16. $\int \frac{1}{a^2+x^2}dx=\frac{1}{a}arctan\frac{x}{a}+C(a>0)$
17. $\int \frac{1}{\sqrt{x^2+a^2}}dx=In(x+\sqrt{x^2+a^2})+C$
18. $\int \frac{1}{\sqrt{x^2-a^2}}dx=In(x+\sqrt{x^2-a^2})+C(\mid x\mid >\mid a\mid )$
19. $\int \frac{1}{x^2-a^2}dx=\frac{1}{2a}In\mid \frac{x-a}{x+a}\mid +C$
20. $\int \frac{1}{a^2-x^2}dx=\frac{1}{2a}In\mid \frac{x+a}{x-a}\mid +C$
21. $\int \sqrt{a^2-x^2}dx=\frac{a^2}{2}arcsin\frac{x}{a}+\frac{x}{2}\sqrt{a^2-x^2}+C(\mid x\mid <a)$
22. $\int si{n}^{2}xdx=\frac{x}{2}-\frac{sin2x}{4}+C$
23. $\int co{s}^{2}xdx=\frac{x}{2}+\frac{sin2x}{4}+C$

7.3 Common integral formula

1. ${\int }_a^{b}f(x)dx={\int }_a^{b}f(a+b-x)dx$
2. ${\int }_a^{b}f(x)dx=\frac{1}{2}{\int }_a^b[f(x)+f(a+b-x)]dx$
3. ${\int }_a^{b}f(x)dx={\int }_a^{\frac{a+b}{2}}[f(x)+f(a+b-x)]dx$
4. Ignition formula ：${\int }_0^{\frac{\pi }{2}}si{n}^{n}xdx={\int }_0^{\frac{\pi }{2}}co{s}^{n}xdx=\frac{n-1}{n}\frac{n-3}{n-2}\cdots$
5. ${\int }_0^{\pi }xf(sinx)dx=\frac{\pi }{2}{\int }_0^{\pi }f(sinx)dx=\pi {\int }_0^{\frac{\pi }{2}}f(sinx)$
6. ${\int }_0^{\frac{\pi }{2}}f(sinx)dx={\int }_0^{\frac{\pi }{2}}f(cosx)dx$
7. ${\int }_0^{\frac{\pi }{2}}f(sinx,cosx)dx={\int }_0^{\frac{\pi }{2}}f(cosx,sinx)dx$
8. ${\int }_a^{b}f(x)dx={\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}f(\frac{a+b}{2}+\frac{b-a}{2}sint)\cdot \frac{b-a}{2}costdt,(x-\frac{a+b}{2}=\frac{b-a}{2}sint)$
9. ${\int }_a^{b}f(x)dx={\int }_0^1(b-a)f[a+(b-a)t]dt,(x-a=(b-a)t)$
10. ${\int }_{-a}^{a}f(x)dx={\int }_0^a[f(x)+f(-x)]dx$
11. ${\int }_0^{n\pi }x\mid sinx\mid dx=n^2\pi$

7.4 Common methods of integration

Make a difference 、 Exchange element 、 Partial integral 、 General points

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