前言

求导容易积分难 🙁

求导

${\left ( sin(x)\right )}’=cos(x)$
${\left ( cos(x)\right )}’=-sin(x)$
${\left ( tan(x)\right )}’=sec^2(x)$
${\left ( cot(x)\right )}’=-csc^2(x)$
${\left ( sec(x)\right )}’=sec(x)tan(x)$
${\left ( csc(x)\right )}’=-csc(x)cot(x)$
${\left ( arcsin(x)\right )}’=\frac{1}{\sqrt{1-x^2}}$
${\left ( arccos(x)\right )}’=-\frac{1}{\sqrt{1-x^2}}$
${\left ( arctan(x)\right )}’=\frac{1}{1+x^2}$
${\left ( arccot(x)\right )}’=-\frac{1}{1+x^2}$
${\left ( \ln(x+\sqrt{x^2\pm a^2})\right )}’=\frac{1}{\sqrt{x^2\pm a^2}}$

积分

$\int \frac{dx}{a^2+x^2}=\frac{1}{a}arctan(\frac{x}{a})+C$
$\int \frac{dx}{a^2-x^2}=\frac{1}{2a}\ln |\frac{a+x}{a-x}|+C$
$\int \frac{dx}{x^2-a^2}=\frac{1}{2a}\ln |\frac{x-a}{x+a}|+C$
$\int \frac{dx}{\sqrt{a^2-x^2}}=arcsin(\frac{x}{a})+C$
$\int \frac{dx}{\sqrt{x^2\pm a^2}}=\ln (x+\sqrt{x^2\pm a^2})+C$
$\int secxdx=\ln \left | secx+tanx \right |+C$
$\int cscxdx=\ln \left | cscx-cotx \right |+C$
$\int \sqrt{x^2+a^2}dx=\frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}\ln(x+\sqrt{x^2+a^2})+C$
$\int \sqrt{x^2-a^2}dx=\frac{x}{2}\sqrt{x^2-a^2}-\frac{a^2}{2}\ln|x+\sqrt{x^2-a^2}|+C$
$\int \sqrt{a^2-x^2}dx=\frac{x}{2}\sqrt{a^2-x^2}+\frac{a^2}{2}arcsin(\frac{x}{a})+C$

勘误