$$\begin{array}{rlrlrl}\int \frac{1}{a^2+x^2}dx& =\frac{1}{a}\arctan \frac{x}{a}+C(a>0)& \int \frac{1}{a^2?x^2}dx& =\frac{1}{2a}\ln |\frac{x+a}{x?a}|+C& \int \frac{1}{x^2?a^2}dx& =\frac{1}{2a}\ln |\frac{x?a}{x+a}|+C\\ \int \frac{1}{\sqrt{x^2+a^2}}dx& =\ln (x+\sqrt{x^2+a^2})+C& \int \frac{1}{\sqrt{a^2?x^2}}dx& =\arcsin \frac{x}{a}+C(a>0)& \int \frac{1}{\sqrt{x^2?a^2}}dx& =\ln (x+\sqrt{x^2?a^2})+C(|x|>|a|)\\ \int \sqrt{a^2+x^2}dx& =& \int \sqrt{a^2?x^2}dx& =\frac{a^2}{2}\arcsin \frac{x}{a}+\frac{x}{2}\sqrt{a^2?x^2}+C& \int \sqrt{x^2?a^2}dx& =\end{array}$$

$$\begin{array}{rl}& (\ln |\sec x+\tan x|+C{)}^{\prime \prime }=(\sec x{)}^{\prime }=\sec x\tan x\\ & (\ln |\csc x?\cot x|+C{)}^{\prime \prime }=(\csc x{)}^{\prime }=?\csc x\cot x\\ & (\ln |\cos x|+C{)}^{\prime \prime }=(?\tan x{)}^{\prime }=?{\sec }^{2}x\\ & (\ln |\sin x|+C{)}^{\prime \prime }=(\cot x{)}^{\prime }=?{\csc }^{2}x\end{array}$$

總結

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