document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=363011>Question 363011</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please help me solve this equation:Use the Comparison Theorem to determine whether the following integrals are<BR>\n" );
document.write( "convergent or divergent.\r<BR>\n" );
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document.write( "a.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+int+%28+root%283%2Cx%29%2F%28sqrt%28x%29-pi%29%2C+dx%2C+1%2C+infinity+%29+ ALIGN=MIDDLE  ALT=\"+int+%28+root%283%2Cx%29%2F%28sqrt%28x%29-pi%29%2C+dx%2C+1%2C+infinity+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "b.  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+int+%28+e%5E%28-x%5E2%29%2C+dx%2C+1%2C+infinity+%29+ ALIGN=MIDDLE  ALT=\"+int+%28+e%5E%28-x%5E2%29%2C+dx%2C+1%2C+infinity+%29+\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #258759 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=258759\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a. Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w+=+sqrt%28x%29-pi ALIGN=MIDDLE  ALT=\"w+=+sqrt%28x%29-pi\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28w%2Bpi%29%5E2 ALIGN=MIDDLE  ALT=\"x+=+%28w%2Bpi%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dx+=+2sqrt%28x%29dw ALIGN=MIDDLE  ALT=\"dx+=+2sqrt%28x%29dw\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Then<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+%28root%283%2Cx%29%2F%28sqrt%28x%29-pi%29%29dx++=+%28%28w%2Bpi%29%5E%285%2F3%29%2Fw%29dw ALIGN=MIDDLE  ALT=\"+%28root%283%2Cx%29%2F%28sqrt%28x%29-pi%29%29dx++=+%28%28w%2Bpi%29%5E%285%2F3%29%2Fw%29dw\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Now <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28w%2Bpi%29%5E%285%2F3%29%2Fw+%3E+=+w%5E%285%2F3%29%2Fw++=++w%5E%282%2F3%29 ALIGN=MIDDLE  ALT=\"%28w%2Bpi%29%5E%285%2F3%29%2Fw+%3E+=+w%5E%285%2F3%29%2Fw++=++w%5E%282%2F3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  It is enough to check the integral for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E=pi%5E2 ALIGN=MIDDLE  ALT=\"x%3E=pi%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > for divergence/convergence (Why?).  We have then  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+int+%28root%283%2Cx%29%2F%28sqrt%28x%29-pi%29%2C+dx%2C+pi%5E2%2C+infinity+%29+=+int%28%28w%2Bpi%29%5E%285%2F3%29%2Fw%2C+dw%2C+0%2C+infinity%29+%3E=+int%28w%5E%282%2F3%29%2C+dw%2C+0%2Cinfinity%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "But the last integral has infinite value, and thus, the given integral itself has infinite value.  The integral is divergent.\r<BR>\n" );
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document.write( "b.<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=+int+%28+e%5E%28-x%5E2%29%2C+dx%2C+1%2C+infinity+%29%3C=+int%28e%5E%28-x%29%2Cdx%2C+1%2C+infinity%29+=+1%2Fe+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.  Therefore the given integral is convergent. </SPAN>\n" );
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