📄 第三节 用mathematica作积分计算.htm
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<title>第三节 用Mathematica作积分计算</title>
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<P align=center class=MsoNormal style="text-align: center; line-height: 200%"><span style="font-family: 宋体; mso-hansi-font-family: Times New Roman; mso-ascii-font-family: Times New Roman"><a href="index.htm">主目录</a></span><span style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">
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style="text-align: center; text-indent: 17.95pt; line-height: 200%"> </P>
<P align=center class=MsoNormal
style="text-align: center; text-indent: 17.95pt; line-height: 200%"><font color="#FF0000" size="6"><b><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">第三节</SPAN>
<SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">用</SPAN><SPAN
lang=EN-US>Mathematica</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">作积分计算</SPAN></b></font></P>
<P class=MsoNormal style="text-indent: 17.95pt; line-height: 200%"><font size="4"><b><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">积分学包括不定积分和定积分两大部分，它与微分学有着密切的联系。微积分的基本定理又称牛顿</SPAN><SPAN
lang=EN-US>-</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">莱布尼兹（</SPAN><SPAN
lang=EN-US>Newton-Leibniz</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">）公式，它揭示了微分与积分之间本质的联系，即微分与积分是互逆的运算，同时揭示了它与不定积分之间的联系，即积分与不定积分都归结为求原函数。积分在几何、物理等方面有着广泛的应用。</SPAN><SPAN
lang=EN-US>Mathematica</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">软件包中，积分主要是通过函数</SPAN><SPAN
lang=EN-US>Integrate[ ]</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">来完成，它包括不定积分和定积分。本节将通过大量的实例来讲解</SPAN><SPAN
lang=EN-US>Mathematica4.0</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">中积分函数的使用及注意事项。</SPAN></b></font></P>
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<P class=MsoNormal style="line-height: 200%"><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'"><b><font size="5" color="#FF0000">一、<a name="不定积分">不定积分</a></font></b></SPAN></P>
<P class=MsoNormal style="text-indent: 17.95pt; line-height: 200%"><font size="4"><b><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">函数</SPAN><SPAN
lang=EN-US>Integrate[f,x]</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">和</SPAN><SPAN
lang=EN-US>D[f,x]</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">为互逆的运算，如果对一个函数</SPAN><SPAN
lang=EN-US>f</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">先做积分运算</SPAN><SPAN
lang=EN-US>Integrate[f,x]</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">再求导</SPAN><SPAN
lang=EN-US>D[f,x]</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">，则会得到函数</SPAN><SPAN
lang=EN-US>f</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">本身。另外，不定积分的计算结果并不输出常数</SPAN><SPAN
lang=EN-US>C</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">，观察如下运算：</SPAN></b></font></P>
<P class=MsoNormal style="text-indent: 17.95pt; line-height: 200%"><SPAN
lang=EN-US><font size="4"><b>In[1]:=Integrate[Cos[x], x]</b></font></SPAN></P>
<P class=MsoNormal style="text-indent: 17.95pt; line-height: 200%"><SPAN
lang=EN-US><font size="4"><b>Out[1]=Sin[x]</b></font></SPAN></P>
<P class=MsoNormal style="text-indent: 17.95pt; line-height: 200%"><SPAN lang=EN-US><font size="4"><b>In[2]:=D[Sin[x]
+ c, x]</b></font></SPAN></P>
<P class=MsoNormal style="text-indent: 17.95pt; line-height: 200%"><SPAN
lang=EN-US><font size="4"><b>Out[2]=Cos[x]</b></font></SPAN></P>
<P class=MsoNormal style="text-indent: 17.95pt; line-height: 200%"><font size="4"><b><SPAN
lang=EN-US>Mathematica4.0</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">在做积分运算时作了如下<font color="#FF0000">约定：</font></SPAN></b></font></P>
<P class=MsoNormal style="line-height: 200%"><font size="4"><b><font color="#FF0000"><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">（</SPAN><SPAN
lang=EN-US>1</SPAN></font><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'"><font color="#FF0000">）</font>对于与积分变量无关的变量，</SPAN><SPAN
lang=EN-US>Integrate</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">函数总是假定它和积分变量相互独立，计算时把它当作常数对待。例如：</SPAN><SPAN
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eqn="prod @3 21600 pixelWidth"><V:F eqn="prod @3 21600 pixelHeight"><V:F
eqn="sum @0 0 1"><V:F eqn="prod @6 1 2"><V:F eqn="prod @7 21600 pixelWidth"><V:F
eqn="sum @8 21600 0"><V:F eqn="prod @7 21600 pixelHeight"><V:F
eqn="sum @10 21600 0"></V:FORMULAS><V:PATH o:connecttype="rect"
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v:ext="edit"></V:SHAPETYPE><V:SHAPE id=_x0000_i1025
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</SPAN></b></font></P>
<P class=MsoNormal style="text-indent: 17.95pt; line-height: 200%"><SPAN
lang=EN-US><font size="4"><b>In[3]:=Integrate[a*Sin[x] + b*Cos[x], x]</b></font></SPAN></P>
<P class=MsoNormal style="text-indent: 17.95pt; line-height: 200%"><SPAN lang=EN-US><font size="4"><b>Out[3]=-a
Cos[x] + b Sin[x]</b></font></SPAN></P>
<P class=MsoNormal style="text-indent: 17.95pt; line-height: 200%"><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'"><b><font size="4">上述计算中</font></b></SPAN><b><font size="4"><SPAN
lang=EN-US>Mathematica4.0</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">把</SPAN><SPAN
lang=EN-US>a</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">和</SPAN><SPAN
lang=EN-US>b</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">当作常数。</SPAN></font></b></P>
<P class=MsoNormal style="line-height: 200%"><font size="4"><b><font color="#FF0000"><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">（</SPAN><SPAN
lang=EN-US>2</SPAN></font><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'"><font color="#FF0000">）</font>对变量的运算只取一般情况，不考虑特殊值。例如：</SPAN><SPAN
lang=EN-US><SPAN style="mso-text-raise: -8.0pt"><!--[if gte vml 1]><V:SHAPE
id=_x0000_i1026 style="HEIGHT: 21.75pt; WIDTH: 33.75pt" o:ole=""
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</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">的运算结果是</SPAN><SPAN
lang=EN-US><SPAN style="mso-text-raise: -12.0pt"><!--[if gte vml 1]><V:SHAPE
id=_x0000_i1027 style="HEIGHT: 33pt; WIDTH: 27pt" o:ole="" type="#_x0000_t75">
<V:IMAGEDATA o:title=""
src="file:///C:/WINDOWS/TEMP/msoclip1/01/clip_image005.wmz"></V:SHAPE><![endif]--><IMG src="images/3.htm75.gif"
v:shapes="_x0000_i1027" width="36" height="44"></SPAN><!--[if gte mso 9]><xml>
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</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">，显然在取特殊值</SPAN><SPAN
lang=EN-US>n=-1</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">时不成立。比较：</SPAN></b></font></P>
<P class=MsoNormal style="text-indent: 17.95pt; line-height: 200%"><SPAN
lang=EN-US><font size="4"><b>In[4]:=Integrate[x^n, x]</b></font></SPAN></P>
<P class=MsoNormal style="text-indent: 17.95pt; line-height: 200%"><SPAN lang=EN-US><font size="4"><b>Out[4]= <SPAN
style="mso-text-raise: -12.0pt"><!--[if gte vml 1]><V:SHAPE id=_x0000_i1028
style="HEIGHT: 33pt; WIDTH: 27pt" o:ole="" type="#_x0000_t75"><V:IMAGEDATA
o:title=""
src="file:///C:/WINDOWS/TEMP/msoclip1/01/clip_image007.wmz"></V:SHAPE><![endif]--><IMG src="images/3.htm76.gif"
v:shapes="_x0000_i1028" width="36" height="44"></SPAN><!--[if gte mso 9]><xml>
<o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1028"
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</b></font>
</SPAN></P>
<P class=MsoNormal style="text-indent: 17.95pt; line-height: 200%"><SPAN
lang=EN-US><font size="4"><b>In[5]:=Integrate[x^(-1), x]</b></font></SPAN></P>
<P class=MsoNormal style="text-indent: 17.95pt; line-height: 200%"><SPAN
lang=EN-US><font size="4"><b>Out[5]=Log[x]</b></font></SPAN></P>
<P class=MsoNormal style="line-height: 200%"><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'"><font color="#FF0000"><b><font size="4">（</font></b></font></SPAN><b><font size="4"><font color="#FF0000"><SPAN
lang=EN-US>3</SPAN></font><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'"><font color="#FF0000">）</font>由于积分相对于微分来说是一个比较困难的过程。微分有一套系统的生成规则，积分没有统一的方式计算。微分运算时，结果与要微分的函数是同一种类型函数，至少是相似的。</SPAN><SPAN
lang=EN-US>Mathematica</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">做积分运算时，结果总是很复杂，它尽量保持与要积分的函数是同一类型。比较：</SPAN></font></b></P>
<P class=MsoNormal style="text-indent: 17.95pt; line-height: 200%"><SPAN
lang=EN-US><font size="4"><b>In[6]:=Integrate[Log[x], x]</b></font></SPAN></P>
<P class=MsoNormal style="text-indent: 17.95pt; line-height: 200%"><SPAN lang=EN-US><font size="4"><b>Out[6]=-x + x
Log[x]</b></font></SPAN></P>
<P class=MsoNormal style="text-indent: 17.95pt; line-height: 200%"><SPAN
lang=EN-US><font size="4"><b>In[7]:=Integrate[Log[Log[x]], x]</b></font></SPAN></P>
<P class=MsoNormal style="text-indent: 17.95pt; line-height: 200%"><SPAN lang=EN-US><font size="4"><b>Out[7]=x
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