📄 第三节 用mathematica作积分计算.htm
字号:
Log[Log[x]] - LogIntegral[x]</b></font></SPAN></P>
<P class=MsoNormal style="text-indent: 17.95pt; line-height: 200%"><font size="4"><b><SPAN
lang=EN-US>LogIntegral</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">为内部函数，其定义为：</SPAN><SPAN
lang=EN-US><SPAN style="mso-text-raise: -9.0pt"><!--[if gte vml 1]><V:SHAPE
id=_x0000_i1029 style="HEIGHT: 26.25pt; WIDTH: 54.75pt" o:ole=""
type="#_x0000_t75"> <V:IMAGEDATA o:title=""
src="file:///C:/WINDOWS/TEMP/msoclip1/01/clip_image009.wmz"></V:SHAPE><![endif]--><IMG src="images/3.htm77.gif"
v:shapes="_x0000_i1029" width="73" height="35"></SPAN><!--[if gte mso 9]><xml>
<o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1029"
DrawAspect="Content" ObjectID="_1048959437">
</o:OLEObject>
</xml><![endif]-->
</SPAN><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>z>1</SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">）。</SPAN></b></font></P>
<P class=MsoNormal style="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'">例</SPAN><SPAN
lang=EN-US>1 </SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">求下列不定积分：</SPAN></b></font></P>
<P class=MsoNormal style="text-indent: 17.95pt; line-height: 200%"><SPAN lang=EN-US><font size="4"><b><SPAN
style="mso-text-raise: -54.0pt"><!--[if gte vml 1]><V:SHAPE id=_x0000_i1030
style="HEIGHT: 107.25pt; WIDTH: 242.25pt" o:ole=""
type="#_x0000_t75"><V:IMAGEDATA o:title=""
src="file:///C:/WINDOWS/TEMP/msoclip1/01/clip_image011.wmz"></V:SHAPE><![endif]--><IMG src="images/3.htm78.gif"
v:shapes="_x0000_i1030" width="323" height="143"></SPAN><!--[if gte mso 9]><xml>
<o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1030"
DrawAspect="Content" ObjectID="_1048959438">
</o:OLEObject>
</xml><![endif]-->
</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'"><font size="4"><b>解：</b></font></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'">（</SPAN><SPAN
lang=EN-US>1</SPAN><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>In[1]:=Integrate[(3 - x^2)^3, x]</SPAN></b></font></P>
<P class=MsoNormal style="text-indent: 17.95pt; line-height: 200%"><SPAN lang=EN-US><font size="4"><b>
Out[1]=<SPAN
style="mso-text-raise: -12.0pt"><!--[if gte vml 1]><V:SHAPE id=_x0000_i1031
style="HEIGHT: 33pt; WIDTH: 108pt" o:ole="" type="#_x0000_t75"> <V:IMAGEDATA
o:title=""
src="file:///C:/WINDOWS/TEMP/msoclip1/01/clip_image013.wmz"></V:SHAPE><![endif]--><IMG src="images/3.htm79.gif"
v:shapes="_x0000_i1031" width="144" height="44"></SPAN><!--[if gte mso 9]><xml>
<o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1031"
DrawAspect="Content" ObjectID="_1048959439">
</o:OLEObject>
</xml><![endif]-->
</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">（</font></b></SPAN><b><font size="4"><SPAN
lang=EN-US>2</SPAN><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>In[2]:=Integrate[x^2/(1 + x^2), x]</SPAN></font></b></P>
<P class=MsoNormal style="text-indent: 17.95pt; line-height: 200%"><SPAN lang=EN-US><font size="4"><b>
Out[2]=x -
ArcTan[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">（</font></b></SPAN><b><font size="4"><SPAN
lang=EN-US>3</SPAN><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>In[3]:=Integrate[(Cot[x])^2, x]</SPAN></font></b></P>
<P class=MsoNormal style="text-indent: 17.95pt; line-height: 200%"><SPAN lang=EN-US><font size="4"><b>
Out[3]=-x -
Cot[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">（</font></b></SPAN><b><font size="4"><SPAN
lang=EN-US>4</SPAN><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>In[4]:=Integrate[x/(1 + x^2 + (1 + x^2)^(3/2)), x]</SPAN></font></b></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_i1032
style="HEIGHT: 33pt; WIDTH: 221.25pt" o:ole="" type="#_x0000_t75"> <V:IMAGEDATA
o:title=""
src="file:///C:/WINDOWS/TEMP/msoclip1/01/clip_image015.wmz"></V:SHAPE><![endif]--><IMG src="images/3.htm80.gif"
v:shapes="_x0000_i1032" width="295" height="44"></SPAN><!--[if gte mso 9]><xml>
<o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1032"
DrawAspect="Content" ObjectID="_1048959440">
</o:OLEObject>
</xml><![endif]-->
</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">（</font></b></SPAN><b><font size="4"><SPAN
lang=EN-US>5</SPAN><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>In[5]:=Integrate[Sin[x]Cos[x + a], x]</SPAN></font></b></P>
<P class=MsoNormal style="text-indent: 17.95pt; line-height: 200%"><SPAN lang=EN-US><font size="4"><b>
Out[5]=<SPAN
style="mso-text-raise: -12.0pt"><!--[if gte vml 1]><V:SHAPE id=_x0000_i1033
style="HEIGHT: 30.75pt; WIDTH: 249pt" o:ole="" type="#_x0000_t75"> <V:IMAGEDATA
o:title=""
src="file:///C:/WINDOWS/TEMP/msoclip1/01/clip_image017.wmz"></V:SHAPE><![endif]--><IMG src="images/3.htm81.gif"
v:shapes="_x0000_i1033" width="332" height="41"></SPAN><!--[if gte mso 9]><xml>
<o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1033"
DrawAspect="Content" ObjectID="_1048959441">
</o:OLEObject>
</xml><![endif]-->
</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">（</font></b></SPAN><b><font size="4"><SPAN
lang=EN-US>6</SPAN><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>In[6]:=Integrate[Log[Sin[x]]/(Sin[x])^2, x]</SPAN></font></b></P>
<P class=MsoNormal style="text-indent: 17.95pt; line-height: 200%"><SPAN lang=EN-US><font size="4"><b>
Out[6]=-x -
Cot[x] - Cot[x] Log[Sin[x]]</b></font>
</SPAN></P>
<P class=MsoNormal style="text-indent: 17.95pt; line-height: 200%"> </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'"><b><font size="5" color="#FF0000">二、<a name="定积分">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'">定积分使用的函数仍然是</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'">，只是多了几个参数而已。以下为定积分函数</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'">的使用格式：</SPAN></b></font></P>
<P class=MsoNormal style="text-indent: 17.95pt; line-height: 200%"><SPAN lang=EN-US><b><font size="4">Integrate[f
</font></b>
</SPAN><b><font size="4"><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>{x</SPAN><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>xmin</SPAN><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>xmax}]<SPAN
style="mso-spacerun: yes"> </SPAN></SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">定积分</SPAN><SPAN
lang=EN-US><SPAN style="mso-text-raise: -9.0pt"><!--[if gte vml 1]><V:SHAPE
id=_x0000_i1034 style="HEIGHT: 26.25pt; WIDTH: 45pt" o:ole=""
type="#_x0000_t75"> <V:IMAGEDATA o:title=""
src="file:///C:/WINDOWS/TEMP/msoclip1/01/clip_image019.wmz"></V:SHAPE><![endif]--><IMG src="images/3.htm82.gif"
v:shapes="_x0000_i1034" width="60" height="35"></SPAN><!--[if gte mso 9]><xml>
<o:OLEObject Type="Embed" ProgID="Equation.3" ShapeID="_x0000_i1034"
DrawAspect="Content" ObjectID="_1048959442">
</o:OLEObject>
</xml><![endif]-->
</SPAN></font></b></P>
<P class=MsoNormal style="text-indent: 17.95pt; line-height: 200%"><SPAN lang=EN-US><b><font size="4">Integrate[f
</font></b>
</SPAN><b><font size="4"><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>{x</SPAN><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>xmin</SPAN><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>xmax}</SPAN><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>{y</SPAN><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>ymin</SPAN><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>ymax}]<SPAN style="mso-spacerun: yes"> <br>
</SPAN></SPAN><SPAN
style="FONT-FAMILY: 宋体; mso-hansi-font-family: 'Times New Roman'; mso-ascii-font-family: 'Times New Roman'">多重积分积分</SPAN><SPAN
lang=EN-US><SPAN style="mso-text-raise: -11.0pt"><!--[if gte vml 1]><V:SHAPE
id=_x0000_i1035 style="HEIGHT: 27.75pt; WIDTH: 81.75pt" o:ole=""
type="#_x0000_t75"> <V:IMAGEDATA o:title=""
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