limx→∞f(x)x=k,limx→∞[f(x)−kx]=b\lim_{x\to \infty}\frac{f(x)}{x}=k,\lim_{x\to\infty}[f(x)-kx]=bx→∞limxf(x)=k,x→∞lim[f(x)−kx]=b
limx→∞f(x)=A\lim_{x \to \infty}f(x)=Ax→∞limf(x)=A
limx→af(x)=∞\lim_{x \to a}f(x)=\inftyx→alimf(x)=∞
∫dxx2+a2=ln(x+x2+a2)+C\int\frac{dx}{\sqrt{x^2+a^2}}=\ln(x+\sqrt{x^2+a^2})+C∫x2+a2dx=ln(x+x2+a2)+C
∫dxx2−a2=ln(x+x2−a2)+C\int\frac{dx}{\sqrt{x^2-a^2}}=\ln(x+\sqrt{x^2-a^2})+C∫x2−a2dx=ln(x+x2−a2)+C
∫01f(x)=limn→∞1n∑i=1nf(in)\int_0^1f(x)=\lim_{n \to \infty}\frac{1}{n}\sum_{i=1}^{n}f(\frac{i}{n})∫01f(x)=n→∞limn1i=1∑nf(ni)
1+tan2x=sec2x1+tan^2x=sec^2x1+tan2x=sec2x
dtanx=sec2xdxdtanx=sec^2xdxdtanx=sec2xdx
dsecx=secxtanxdxdsecx=secxtanxdxdsecx=secxtanxdx
∫secxdx=ln∣secx+tanx∣+C\int secx dx=ln|secx+tanx|+C∫secxdx=ln∣secx+tanx∣+C
∫tanxdx=ln∣cosx∣+C\int tanxdx=ln|cosx|+C ∫tanxdx=ln∣cosx∣+C
ex=1+x+x22!+x33!+...+xnn!+o(xn)e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...+\frac{x^n}{n!}+o(x^n) ex=1+x+2!x2+3!x3+...+n!xn+o(xn)
ln(1+x)=x−x22!+x33!−...+(−1)n−1xnn!ln(1+x)=x-\frac{x^2}{2!}+\frac{x^3}{3!}-...+\frac{(-1)^{n-1}x^n}{n!}ln(1+x)=x−2!x2+3!x3−...+n!(−1)n−1xn
tanxtanxtanx
sinx=x−x33!+x55!−...+−(−1)nx2n+1(2n+1)!+o(x2n+1)sinx=x-\frac{x^3}{3!}+\frac{x^5}{5!}-...+-\frac{(-1)^nx^{2n+1}}{(2n+1)!}+o(x^{2n+1})sinx=x−3!x3+5!x5−...+−(2n+1)!(−1)nx2n+1+o(x2n+1)
cosx=1−x22!+x44!−...+(−1)nx2n2n!+o(x2n+1)cosx=1-\frac{x^2}{2!}+\frac{x^4}{4!}-...+\frac{(-1)^nx^{2n}}{2n!}+o(x^{2n+1})cosx=1−2!x2+4!x4−...+2n!(−1)nx2n+o(x2n+1)
(1+x)a=1+ax+a(a−1)2!x2+a(a−1)(a−2)23!x3+...+o(xn)(1+x)^a=1+ax+\frac{a(a-1)}{2!}x^2+\frac{a(a-1)(a-2)^2}{3!}x^3+...+o(x^n)(1+x)a=1+ax+2!a(a−1)x2+3!a(a−1)(a−2)2x3+...+o(xn)
11−x=1+x+x2+...+xn+o(xn)\frac{1}{1-x}=1+x+x^2+...+x^n+o(x^n)1−x1=1+x+x2+...+xn+o(xn)
11+x=1−x+x2−...+(−1)nxn+o(xn)\frac{1}{1+x}=1-x+x^2-...+(-1)^nx^n+o(x^n)1+x1=1−x+x2−...+(−1)nxn+o(xn)
Φ(x)=∫0xf(t)dt,则Φ′(x)=f(x)\Phi(x)=\int_0^xf(t)dt,则\Phi '(x)=f(x)Φ(x)=∫0xf(t)dt,则Φ′(x)=f(x)
∫0π2sinnxdx=∫0π2cosnxdx=In=n−1nIn−2\int_0^{\frac{\pi}{2}}sin^nxdx=\int_0^{\frac{\pi}{2}}cos^nxdx=I_n=\frac{n-1}{n}I_{n-2}∫02πsinnxdx=∫02πcosnxdx=In=nn−1In−2