document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=407767>Question 407767</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Evaluate the integral using integration by parts with explanations.\r<BR>\n" );
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document.write( "ln (2x+1) dx </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #287367 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=287367\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is an alternative approach which can still use integration by parts, but is slightly easier.\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u+=+2x%2B1 ALIGN=MIDDLE  ALT=\"u+=+2x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=du+=+2dx ALIGN=MIDDLE  ALT=\"du+=+2dx\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dx+=+du%2F2 ALIGN=MIDDLE  ALT=\"dx+=+du%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" >. If we replace 2x+1, we obtain\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29int%28ln%28u%29%2C+du%29 ALIGN=MIDDLE  ALT=\"%281%2F2%29int%28ln%28u%29%2C+du%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. We can integrate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28ln%28u%29%2C+du%29 ALIGN=MIDDLE  ALT=\"int%28ln%28u%29%2C+du%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > via parts, or by a formula which you can (and should) memorize.\r<BR>\n" );
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document.write( "Letting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+ln+%28u%29 ALIGN=MIDDLE  ALT=\"y+=+ln+%28u%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dz+=+du ALIGN=MIDDLE  ALT=\"dz+=+du\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we obtain <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy+=+du%2Fu ALIGN=MIDDLE  ALT=\"dy+=+du%2Fu\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=z+=+u ALIGN=MIDDLE  ALT=\"z+=+u\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Applying the parts formula,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28ln%28u%29%2C+du%29+=+u%2Aln+%28u%29+-+int%28u%2Fu%2C+du%29+=+u%2Aln+%28u%29+-+u+%2B+C ALIGN=MIDDLE  ALT=\"int%28ln%28u%29%2C+du%29+=+u%2Aln+%28u%29+-+int%28u%2Fu%2C+du%29+=+u%2Aln+%28u%29+-+u+%2B+C\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Note that we have to multiply by 1/2, since we want <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29int%28ln%28u%29%2C+du%29 ALIGN=MIDDLE  ALT=\"%281%2F2%29int%28ln%28u%29%2C+du%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Back-substitute <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u+=+2x%2B1 ALIGN=MIDDLE  ALT=\"u+=+2x%2B1\"   DISABLED_style= \"max-width:90%; height:auto;\" > to obtain\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F2%29int%28ln+%282x%2B1%29%2C+dx%29+=+%281%2F2%29%28%282x%2B1%29%2Aln%282x%2B1%29+-+%282x%2B1%29%29+%2B+C ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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