常见积分求导公式表--便于记忆
∫1a2+x2dx=1aarctan?xa+C(a>0)∫1a2?x2dx=12aln?∣x+ax?a∣+C∫1x2?a2dx=12aln?∣x?ax+a∣+C∫1x2+a2dx=ln?(x+x2+a2)+C∫1a2?x2dx=arcsin?xa+C(a>0)∫1x2?a2dx=ln?(x+x2?a2)+C(∣x∣>∣a∣)∫a2+x2dx=∫a2?x2dx=a22arcsin?xa+x2a2?x2+C∫x2?a2dx=\large \begin{aligned} \int \frac{1}{a^2+x^2}dx&=\frac1a\arctan\frac xa+C(a>0)\qquad&\int\frac{1}{a^2-x^2}dx&=\frac{1}{2a}\ln|\frac{x+a}{x-a}|+C\qquad& \int \frac{1}{x^2-a^2}dx&=\frac{1}{2a}\ln|\frac{x-a}{x+a}|+C\\ \int \frac{1}{\sqrt{x^2+a^2}}dx&=\ln(x+\sqrt{x^2+a^2})+C\qquad&\int \frac{1}{\sqrt{a^2-x^2}}dx&=\arcsin\frac xa+C(a>0)\qquad& \int \frac{1}{\sqrt{x^2-a^2}}dx&=\ln(x+\sqrt{x^2-a^2})+C(|x|>|a|)\\ \int {\sqrt{a^2+x^2}}dx&=\qquad&\int {\sqrt{a^2-x^2}}dx&=\frac{a^2}{2}\arcsin\frac xa+\frac x2\sqrt{a^2-x^2}+C\qquad& \int \sqrt{x^2-a^2}dx&=\\ \end{aligned} ∫a2+x21?dx∫x2+a2?1?dx∫a2+x2?dx?=a1?arctanax?+C(a>0)=ln(x+x2+a2?)+C=?∫a2?x21?dx∫a2?x2?1?dx∫a2?x2?dx?=2a1?ln∣x?ax+a?∣+C=arcsinax?+C(a>0)=2a2?arcsinax?+2x?a2?x2?+C?∫x2?a21?dx∫x2?a2?1?dx∫x2?a2?dx?=2a1?ln∣x+ax?a?∣+C=ln(x+x2?a2?)+C(∣x∣>∣a∣)=?
(ln?∣sec?x+tan?x∣+C)′′=(sec?x)′=sec?xtan?x(ln?∣csc?x?cot?x∣+C)′′=(csc?x)′=?csc?xcot?x(ln?∣cos?x∣+C)′′=(?tan?x)′=?sec?2x(ln?∣sin?x∣+C)′′=(cot?x)′=?csc?2x\begin{aligned} &\large(\ln|\sec x+\tan x|+C)''=(\sec x)'=\sec x\tan x\\ &\large(\ln|\csc x-\cot x|+C)''=(\csc x)'=-\csc x\cot x\\ &\large(\ln|\cos x|+C)''=(-\tan x)'=-\sec^2 x\\ &\large(\ln|\sin x|+C)''=(\cot x)'=-\csc^2 x \end{aligned} ?(ln∣secx+tanx∣+C)′′=(secx)′=secxtanx(ln∣cscx?cotx∣+C)′′=(cscx)′=?cscxcotx(ln∣cosx∣+C)′′=(?tanx)′=?sec2x(ln∣sinx∣+C)′′=(cotx)′=?csc2x?
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