查看“高等数学”的源代码

== 极限 ==
当<math>x\to 0</math>时，常用的等价无穷小

（1）<math>x\sim \sin x\sim\tan x\sim\arcsin x\sim\arctan x\sim\ln_{}{\left ( 1+x \right ) } \sim e^{x} -1</math>

（2）<math>1-\cos x\sim \frac{x^{2}}{2} ,1-\cos ^{a} x\sim\frac{a}{2} x^{2} </math>

（3）<math>\left ( 1+x \right ) ^{a} -1\sim ax</math>

（4）<math>a^{x} -1\sim x\ln_{}{a} </math>
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!<math>\lim_{\bigtriangleup \to 0} \frac{\sin \bigtriangleup }{\bigtriangleup } =1</math>
!<math>\lim_{\bigtriangleup \to 0} \left ( 1+\bigtriangleup  \right ) ^{\frac{1}{\bigtriangleup } } =e</math>
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== 泰勒中值定理 ==
== 不定积分常用三角函数公式 ==
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|<math>\left ( \sin x \right ) '=\cos x</math>
|<math>\int \cos x\;dx=\sin x+C</math>
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|<math>\left ( \cos x \right ) '=-\sin  x </math>
|<math>\int \sin x\;dx=-\cos  x+C</math>
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|<math>\left ( \tan x \right )'=\sec^2 x </math>
|<math>\int \sec ^{2} x\;dx=\tan x+C</math>
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|<math>\left ( \sec x \right )'=\sec x\,\tan x </math>
|<math>\int \sec x\tan x\;dx=\sec x+C</math>
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|<math>\left ( \cot x \right )'=-\csc^2 x </math>
|<math>\int \csc  ^{2} x\;dx=-\cot x+C</math>
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|<math>\left ( \csc x \right )'=- \csc x\,\cot x </math>
|<math>\int \csc x\cot x\;dx=-\csc x+C</math>
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{| class="wikitable"
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|<math>\int \tan x\;dx=\ln_{}{\left | \sec x+\sec x \right | } +C=\ln_{}{\left | \sec x \right | } +C </math>
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|<math>\int \sec x\;dx=\ln_{}{\left | \sec x+\tan x \right | } +C</math>
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|<math>\int \cot x\;dx=-\ln_{}{\left | \sec x+\sec x \right | }+C =-\ln_{}{\left | \sin x \right | } +C</math>
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|<math>\int \csc x\;dx=-\ln_{}{\left | \csc x+\cot x \right | } +C</math>
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|<math>\int \sec ^{3} x\;dx=\frac{1}{2} \left ( \sec x\tan x+\ln_{}{\left | \sec x+\tan x \right | }  \right ) +C</math>
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|<math>\int \csc ^{3} x\;dx=-\frac{1}{2}\left ( \csc x \cot x +\ln_{}{\left | \csc x+\cot x \right | } \right) +C  </math>
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|<math>\int \frac{dx}{\sqrt{1-x^{2}}}=\arcsin x+C </math>
|<math>\int \frac{dx}{\sqrt{a^{2}-x^{2} } }= \arcsin \frac{x}{a} +C</math>
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|<math>\int \frac{dx}{1+x^{2} } =\arctan x+C</math>
|<math>\int \frac{dx}{a^{2}+ x^{2}} =\frac{1}{a} \arctan \frac{x}{a}+C </math>
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{| class="wikitable"
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|<math>\int \frac{dx}{x^{2}-a^{2} } =\frac{1}{2a} \ln_{}{\left |\frac{x-a}{x+a}  \right | } +C</math>
|<math>\int \frac{dx}{\sqrt{x^{2}\pm a^{2} } } =\ln_{}{\left | x+\sqrt{x^{2}\pm a^{2}} \right | } +C</math>
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{| class="wikitable"
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|<math>\int \sqrt{a^{2}-x^{2} }\;dx=\frac{a^{2} }{2}\arcsin \frac{x}{a}+\frac{x}{2}\sqrt{a^{2}-x^{2} }+C  </math>