数学公式01

$$\underset{x\to \infty }{\lim }\frac{f(x)}{x}=k,\underset{x\to \infty }{\lim }[f(x)-kx]=b$$
$$\underset{x\to \infty }{\lim }f(x)=A$$
$$\underset{x\to a}{\lim }f(x)=\infty$$
$$\int \frac{dx}{\sqrt{x^2+a^2}}=\ln (x+\sqrt{x^2+a^2})+C$$
$$\int \frac{dx}{\sqrt{x^2-a^2}}=\ln (x+\sqrt{x^2-a^2})+C$$
$${\int }_0^{1}f(x)=\underset{n\to \infty }{\lim }\frac{1}{n}\sum _{i=1}^{n}f(\frac{i}{n})$$
$$1+ta{n}^{2}x=se{c}^{2}x$$
$$dtanx=se{c}^{2}xdx$$
$$dsecx=secxtanxdx$$
$$\int secxdx=ln|secx+tanx|+C$$
$$\int tanxdx=ln|cosx|+C$$
$$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...+\frac{x^n}{n!}+o(x^n)$$
$$ln(1+x)=x-\frac{x^2}{2!}+\frac{x^3}{3!}-...+\frac{(-1{)}^{n-1}{x}^n}{n!}$$
$$tanx$$
$$sinx=x-\frac{x^3}{3!}+\frac{x^5}{5!}-...+-\frac{(-1{)}^n{x}^{2n+1}}{(2n+1)!}+o(x^{2n+1})$$
$$cosx=1-\frac{x^2}{2!}+\frac{x^4}{4!}-...+\frac{(-1{)}^n{x}^{2n}}{2n!}+o(x^{2n+1})$$
$$(1+x{)}^a=1+ax+\frac{a(a-1)}{2!}{x}^2+\frac{a(a-1)(a-2{)}^2}{3!}{x}^3+...+o(x^n)$$
$$\frac{1}{1-x}=1+x+x^2+...+x^n+o(x^n)$$
$$\frac{1}{1+x}=1-x+x^2-...+(-1{)}^n{x}^n+o(x^n)$$
$$\Phi (x)={\int }_0^{x}f(t)dt,则{\Phi }^{\prime }(x)=f(x)$$
$${\int }_0^{\frac{\pi }{2}}si{n}^{n}xdx={\int }_0^{\frac{\pi }{2}}co{s}^{n}xdx=I_n=\frac{n-1}{n}{I}_{n-2}$$