document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1033411>Question 1033411</A>: <I><SPAN STYLE=\"word-break: break-all;\"> From a Putnam competition:\r<BR>\n" );
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document.write( "Evaluate the integral <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28ln%28x%2B1%29%2F%28x%5E2%2B1%29%2C+dx%2C+0%2C1%29 ALIGN=MIDDLE  ALT=\"int%28ln%28x%2B1%29%2F%28x%5E2%2B1%29%2C+dx%2C+0%2C1%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 #648219 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=648219\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dtan%5E-1%28x%29+=+dx%2F%281%2Bx%5E2%29 ALIGN=MIDDLE  ALT=\"dtan%5E-1%28x%29+=+dx%2F%281%2Bx%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, \r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28ln%28x%2B1%29%2F%28x%5E2%2B1%29%2C+dx%2C+0%2C1%29+=+int%28ln%28x%2B1%29%2C+dtan%5E-1%28x%29%2C+0%2C1%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w+=+tan%5E-1%28x%29 ALIGN=MIDDLE  ALT=\"w+=+tan%5E-1%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  ==> tanw  = x.  Now if x = 0, w = 0, and when x = 1, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=w+=+pi%2F4 ALIGN=MIDDLE  ALT=\"w+=+pi%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "==> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28ln%28x%2B1%29%2F%28x%5E2%2B1%29%2C+dx%2C+0%2C1%29+=+int%28ln%28tanw%2B1%29%2C+dw%2C+0%2Cpi%2F4%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28ln%28tanw%2B1%29%2C+dw%2C+0%2Cpi%2F4%29+=+int%28ln%28%28sinw%2Bcosw%29%2Fcosw%29%2C+dw%2C+0%2Cpi%2F4%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28ln%28sqrt%282%29%2A%28%281%2Fsqrt%282%29%29%2Asinw%2B%281%2Fsqrt%282%29%29%2Acosw%29%29+-+lncosw%2C+dw%2C+0%2Cpi%2F4%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28ln%28sqrt%282%29%29%2B+ln%28sin%28pi%2F4%29%2Asinw%2Bcos%28pi%2F4%29%2Acosw%29+-+lncosw%2C+dw%2C+0%2Cpi%2F4%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28sqrt%282%29%29%2A%28pi%2F4%29%2B+int%28ln%28cos%28pi%2F4+-+w%29%29%2Cdw%2C0%2Cpi%2F4%29+-+int%28ln%28cosw%29%2C+dw%2C+0%2Cpi%2F4%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "But <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28ln%28cos%28pi%2F4+-+w%29%29%2Cdw%2C0%2Cpi%2F4%29+=+int%28ln%28cosw%29%2C+dw%2C+0%2Cpi%2F4%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > by using the substitution <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+pi%2F4+-+w ALIGN=MIDDLE  ALT=\"z+=+pi%2F4+-+w\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the left integral.\r<BR>\n" );
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document.write( "Hence <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28ln%28x%2B1%29%2F%28x%5E2%2B1%29%2C+dx%2C+0%2C1%29++=+ln%28sqrt%282%29%29%2A%28pi%2F4%29+=highlight%28%28pi%2Aln2%29%2F8%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >  </SPAN>\n" );
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