高等数学:修订间差异
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== 极限 == |
== 极限 == |
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{{主条目|[[极限 (数学)|极限]]}} |
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当<math>x\to 0</math>时,常用的等价无穷小 |
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(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> |
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(2)<math>1-\cos x\sim \frac{x^{2}}{2} ,1-\cos ^{a} x\sim\frac{a}{2} x^{2} </math> |
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(3)<math>\left ( 1+x \right ) ^{a} -1\sim ax</math> |
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(4)<math>a^{x} -1\sim x\ln_{}{a} </math> |
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{| class="wikitable" |
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!<math>\lim_{\bigtriangleup \to 0} \frac{\sin \bigtriangleup }{\bigtriangleup } =1</math> |
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!<math>\lim_{\bigtriangleup \to 0} \left ( 1+\bigtriangleup \right ) ^{\frac{1}{\bigtriangleup } } =e</math> |
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|} |
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== 泰勒公式 == |
== 泰勒公式 == |
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{{主条目|[[泰勒公式]]}} |
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如果函数<math>f\left ( x \right )</math>在<math>x=x_{0} </math>的领域内具有n+1阶导数则 |
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<math>f\left(x\right) =f\left( x_{0} \right)+f'\left ( x_{0}\right)\left(x-_{0}\right )+\cdots+\frac{f^{\left(n\right)}\left( x_{0}\right )}{n!}\left(x-x_{0}\right )^{n}+R_{n}\left(x\right ) </math> |
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== 三角函数 == |
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其中<math>\xi </math>介于<math>x</math>与<math>x_{0} </math>之间,<math>R_{n}\left(x\right ) =\frac{f^{\left(n+1\right)}\left( \xi \right)}{\left(n+1\right)!}\left(x-x_{0}\right)^{n+1}</math>称之为拉格朗日余项,余项<math>R_{n}\left(x\right)</math>也可以表示为<math>R_{n}\left(x\right)=o\left(\left(x-x_{0}\right)^{n}\right )</math>。 |
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{{主条目|[[三角函数]]}} |
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(1)当<math>x_{0}=0</math>时,<math>f\left ( x \right ) =f\left ( 0 \right ) +f^{'} \left ( 0 \right ) x+\cdots +\frac{f^{\left(n\right)}\left(0\right)}{n!}\left(x\right)^{n}+R_{n} \left ( x \right ) </math>称为麦克劳林公式。 |
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(2)常用的麦克劳林公式: |
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<math>\ \ \ \ </math><big>①</big><math>\ \ e^{x} =1+x+\frac{x^{2} }{2!} +\cdots +\frac{x^{n} }{n!} + o \left ( x^{n} \right ) </math> |
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<math>\ \ \ \ </math><big>②</big><math>\ \ \sin x=x-\frac{x^{3} }{3!}+\cdots +\frac{\left ( -1 \right )^{n} }{\left ( 2n+1 \right )! } x^{2n+1} + o \left ( x^{2n+1} \right ) </math> |
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<math>\ \ \ \ </math><big>③</big><math>\ \ \cos x=1-\frac{x^{2} }{2!} +\cdots +\frac{\left ( -1 \right )^{n} }{\left ( 2n \right )! } x^{2n}+ o \left ( x^{2n} \right ) </math> |
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<math>\ \ \ \ </math><big>④</big><math>\ \ \frac{1}{1-x} =1+x+x^{2} +\cdots +x^{n} + o \left ( x^{n} \right ) </math> |
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<math>\ \ \ \ </math><big>⑤</big><math>\ \ \frac{1}{1+x}=1-x+x^{2} -\cdots +\left ( -1 \right ) ^{n}x^{n} + o \left ( x^{n} \right ) </math> |
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<math>\ \ \ \ </math><big>⑥</big><math>\ \ \ln_{}{\left ( 1+x \right ) } =x-\frac{x^{2} }{x} +\frac{x^{3} }{3} -\cdots +\frac{\left ( -1 \right ) ^{n-1} }{n}x^{n} + o \left ( x^{n} \right ) </math> |
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<math>\ \ \ \ </math><big>⑦</big><math>\ \ \left ( 1+x \right )^{a} =1+ax+\frac{a\left ( a-1 \right ) }{2!}x^{2}+\cdots +\frac{a\left(a-1\right)\cdots \left(a-n+1\right)}{n!}x^{n} + o \left ( x^{n} \right )</math> |
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<math>\ \ \ \ </math><big>⑧</big><math>\ \ \arctan x=x-\frac{x^{3} }{3} +\frac{x^{5} }{5} -\cdots +\frac{\left ( -1 \right )^{n}}{2n+1} x^{2n+1}+ o \left ( x^{2n+1} \right ) </math> |
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== 三角函数公式 == |
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==== 三角函数 ==== |
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{| class="wikitable" |
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<math>a^{2}+b^{2}=c^{2} </math> |
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|- |
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!正弦(sin) |
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|<math>\sin \alpha =\frac{a}{c}</math> |
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!余割(csc) |
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|<math>\csc \alpha =\frac{c}{a} </math> |
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!<math>\sin \alpha \ \csc \alpha =1</math> |
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|- |
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!余弦(cos) |
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|<math>\cos \alpha =\frac{b}{c}</math> |
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!正割(sec) |
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|<math>\sec \alpha =\frac{c}{b} </math> |
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!<math>\cos \alpha \ \sec \alpha =1</math> |
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|- |
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!正切(tan) |
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|<math>\tan \alpha =\frac{a}{b} </math> |
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!余切(cot) |
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|<math>\cot \alpha =\frac{b}{a} </math> |
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!<math>\tan \alpha \ \cot \alpha =1</math> |
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|} |
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==== 平方关系 ==== |
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{| class="wikitable" |
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|- |
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|<math>\sin ^{2} \alpha +\cos ^{2} \alpha =1</math> |
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|<math>1+\cot ^{2} \alpha =csc^{2} \alpha </math> |
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|<math>\tan ^{2} \alpha +1=\sec ^{2} \alpha </math> |
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|} |
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==== 二角和差公式 ==== |
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{| class="wikitable" |
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|- |
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|<math>\sin \left ( \alpha + \beta \right ) =\sin \alpha \cos \beta+ \cos \alpha\sin \beta</math> |
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|colspan="1" rowspan="2"|<math>\tan \left ( \alpha + \beta \right )=\frac{\tan \alpha +\tan \beta }{1-\tan \alpha \tan \beta } </math> |
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|- |
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|<math>\sin \left ( \alpha - \beta \right ) =\sin \alpha \cos \beta- \cos \alpha\sin \beta</math> |
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|- |
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|<math>\cos \left ( \alpha+\beta \right ) =\cos \alpha \cos \beta -\sin \alpha \sin \beta </math> |
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|colspan="1" rowspan="2"|<math>\tan \left ( \alpha - \beta \right )=\frac{\tan \alpha -\tan \beta }{1+\tan \alpha \tan \beta } </math> |
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|- |
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|<math>\cos \left ( \alpha-\beta \right ) =\cos \alpha \cos \beta+\sin \alpha \sin \beta </math> |
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|} |
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==== 二倍角公式 ==== |
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{| class="wikitable" |
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|- |
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|<math>\sin 2\alpha =\sin \alpha \cos \alpha+\sin \alpha\cos \alpha=2\sin \alpha \cos \alpha </math> |
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|colspan="1" rowspan="2"|<math>\tan2\alpha =\frac{2\tan \alpha }{1-\tan ^{2}\alpha } </math> |
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|- |
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|<math>\cos2 \alpha =\cos ^{2}\alpha -\sin ^{2}\alpha =2\cos ^{2}\alpha -1=1-2\sin ^{2}\alpha</math> |
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|} |
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==== 积化和差公式 ==== |
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{| class="wikitable" |
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|- |
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|<math>\sin \alpha \cos\beta =\frac{1}{2}\left[\sin\left(\alpha+\beta \right)+\sin\left(\alpha -\beta\right)\right] </math> |
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|<math>\cos \alpha \cos \beta =\frac{1}{2} \left [ \cos \left ( \alpha +\beta \right )+\cos \left ( \alpha -\beta \right ) \right ] </math> |
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|- |
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|<math>\cos \alpha \sin \beta =\frac{1}{2} \left [ \sin \left ( \alpha +\beta \right ) -\sin \left ( \alpha -\beta \right ) \right ] </math> |
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|<math>\sin \alpha \ \sin \beta =\frac{1}{2} \left [ \cos \left ( \alpha +\beta \right )-\cos \left ( \alpha -\beta \right ) \right ] </math> |
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|} |
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==== 和差化积公式 ==== |
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{| class="wikitable" |
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|- |
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|<math>\sin \alpha +\sin \beta =2\sin \frac{\alpha +\beta }{2} \cos \frac{\alpha -\beta }{2} </math> |
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|<math>\cos \alpha +\cos \beta =2\cos \frac{\alpha +\beta }{2} \cos\frac{\alpha -\beta }{2} </math> |
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|- |
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|<math>\sin \alpha -\sin \beta =2\cos \frac{\alpha +\beta }{2} \sin \frac{\alpha -\beta }{2} </math> |
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|<math>\cos \alpha -\cos \beta =-2\sin \frac{\alpha +\beta }{2} \sin \frac{\alpha -\beta }{2} </math> |
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|} |
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==== 万能公式 ==== |
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{| class="wikitable" |
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|- |
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|<big><math>\sin \alpha =\frac{2\tan \frac{\alpha }{2} }{1+\tan ^{2}\alpha\frac{\alpha }{2} }</math> |
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|- |
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|<big><math>\cos \alpha =\frac{1-\tan ^{2}\frac{\alpha }{2} }{1+\tan ^{2}\frac{\alpha }{2} } </math> |
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|- |
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|<big><math>\tan \alpha =\frac{2\tan \alpha\frac{\alpha }{2} }{1-\tan ^{2}\frac{\alpha }{2} } </math> |
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|} |
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==== 欧拉公式 ==== |
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{| class="wikitable" |
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|- |
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!<big><math>e^{ix} =\cos x+i\sin x </math> |
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!<big><math>\sin x=\frac{e^{ix}-e^{-ix}}{2i} </math> |
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!<big><math>\cos x=\frac{e^{ix}+e^{-ix}}{2} </math> |
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|} |
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<big>当x=π时,<math>e^{ix} +1=0 </math> |
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== 不定积分常用三角函数公式 == |
== 不定积分常用三角函数公式 == |
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{{主条目|[[不定积分#不定积分常用三角函数公式]]}} |
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{| class="wikitable" |
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|- |
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|<math>\left ( \sin x \right ) '=\cos x</math> |
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|<math>\int \cos x\;dx=\sin x+C</math> |
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|- |
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|<math>\left ( \cos x \right ) '=-\sin x </math> |
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|<math>\int \sin x\;dx=-\cos x+C</math> |
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|- |
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|<math>\left ( \tan x \right )'=\sec^2 x </math> |
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|<math>\int \sec ^{2} x\;dx=\tan x+C</math> |
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|- |
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|<math>\left ( \sec x \right )'=\sec x\,\tan x </math> |
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|<math>\int \sec x\tan x\;dx=\sec x+C</math> |
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|- |
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|<math>\left ( \cot x \right )'=-\csc^2 x </math> |
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|<math>\int \csc ^{2} x\;dx=-\cot x+C</math> |
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|- |
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|<math>\left ( \csc x \right )'=- \csc x\,\cot x </math> |
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|<math>\int \csc x\cot x\;dx=-\csc x+C</math> |
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|} |
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{| class="wikitable" |
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|- |
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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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|- |
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|<math>\int \sec x\;dx=\ln_{}{\left | \sec x+\tan x \right | } +C</math> |
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|- |
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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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|- |
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|<math>\int \csc x\;dx=-\ln_{}{\left | \csc x+\cot x \right | } +C</math> |
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|- |
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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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|- |
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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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|} |
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{| class="wikitable" |
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|- |
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|<math>\int \frac{dx}{\sqrt{1-x^{2}}}=\arcsin x+C </math> |
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|<math>\int \frac{dx}{\sqrt{a^{2}-x^{2} } }= \arcsin \frac{x}{a} +C</math> |
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|- |
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|<math>\int \frac{dx}{1+x^{2} } =\arctan x+C</math> |
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|<math>\int \frac{dx}{a^{2}+ x^{2}} =\frac{1}{a} \arctan \frac{x}{a}+C </math> |
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|} |
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{| class="wikitable" |
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|- |
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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> |
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|<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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|} |
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{| class="wikitable" |
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|- |
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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> |
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== 隐函数求偏导 == |
== 隐函数求偏导 == |
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定理1 设函数1在点1的某个邻域内连续可偏导,且1,1,则在点1的邻域内由1能唯一确定连续可导的函数1,满足1,且 |
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== 微分方程 == |
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1 |
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{{主条目|[[微分方程]]}} |