document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=226356>Question 226356</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi, I was wondering if someone could help me with this question. \r<BR>\n" );
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document.write( "Evaluate the following indefinite integral using integration by parts: <BR>\n" );
document.write( "(ln^(2)(x)) sqrt(x) dx \r<BR>\n" );
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document.write( "Thanks, Judy </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #168593 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=168593\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The basis for integration by parts is the product rule of differentiation:<BR>\n" );
document.write( "d(uv) = u*dv + v*du<BR>\n" );
document.write( "This can be rewritten as:<BR>\n" );
document.write( "u*dv = d(uv) - v*du<BR>\n" );
document.write( "If we integrate this we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28u%2C+dv%29+=+int%28d%28uv%29%2C+dx%29+-+int%28v%2Cdu%29 ALIGN=MIDDLE  ALT=\"int%28u%2C+dv%29+=+int%28d%28uv%29%2C+dx%29+-+int%28v%2Cdu%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28u%2C+dv%29 ALIGN=MIDDLE  ALT=\"int%28u%2C+dv%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = uv - <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28v%2C+du%29 ALIGN=MIDDLE  ALT=\"int%28v%2C+du%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which is the formula for integration by parts. In order to use this formula we are looking to express an integral as a product of some function we'll call \"u\" and the derivative of some other function we'll call \"v\".<BR>\n" );
document.write( "In looking at <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28%28ln%28x%29%29%5E2%2Asqrt%28x%29%2C+dx%29 ALIGN=MIDDLE  ALT=\"int%28%28ln%28x%29%29%5E2%2Asqrt%28x%29%2C+dx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > we want to figure out our \"u\" and \"dv\". Often the key is to pick \"dv\" wisely. Since the other side of the formula has 2 v's, we want \"dv\" to be something easily integrated. Also, since the formula has <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28v%2Adu%29 ALIGN=MIDDLE  ALT=\"int%28v%2Adu%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > on the right, we want v*du to an integral that is easier to find than the one we started with.<BR>\n" );
document.write( "Since powers of x are easily integrable and since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28x%29+=+x%5E%281%2F2%29 ALIGN=MIDDLE  ALT=\"sqrt%28x%29+=+x%5E%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > we will try <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dv+=+x%5E%281%2F2%29dx ALIGN=MIDDLE  ALT=\"dv+=+x%5E%281%2F2%29dx\"   DISABLED_style= \"max-width:90%; height:auto;\" >. This makes <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v+=+%282%2F3%29x%5E%283%2F2%29 ALIGN=MIDDLE  ALT=\"v+=+%282%2F3%29x%5E%283%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u+=+%28ln%28x%29%29%5E2 ALIGN=MIDDLE  ALT=\"u+=+%28ln%28x%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=du+=+2%2Aln%28x%29%2A%281%2Fx%29 ALIGN=MIDDLE  ALT=\"du+=+2%2Aln%28x%29%2A%281%2Fx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Substituting these into our formula:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28%28ln%28x%29%29%5E2%2Asqrt%28x%29%2C+dx%29+=+int%28u%2C+dv%29+=+u%2Av+-+int%28v%2C+du%29+=+%28%28ln%28x%29%29%5E2%29%2A%28%282%2F3%29x%5E%283%2F2%29%29+-+int%28%28%282%2F3%29x%5E%283%2F2%29%29%2A%282%2Aln%28x%29%2A%281%2Fx%29%29+%2C+dx%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%28ln%28x%29%29%5E2%29%2A%28%282%2F3%29x%5E%283%2F2%29%29+-+int%28%28%284%2F3%29x%5E%281%2F2%29%29%2A%28ln%28x%29%29%2C+dx%29+=%0D%0A%28%28ln%28x%29%29%5E2%29%2A%28%282%2F3%29x%5E%283%2F2%29%29+-+%284%2F3%29int%28%28x%5E%281%2F2%29%29%2A%28ln%28x%29%29%2C+dx%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "The integral we still have is not easy. But it is easier than the one we started with. In fact it is very similar to the one we started with. Except it has ln to the first power (instead of squared). Integration by parts can be used to \"reduce\" the complexity of an integral and it can be used repeatedly to reduce the integral until it is one that can be integrated more easily by another method of integration.<BR>\n" );
document.write( "We can use integration by parts on <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28x%5E%281%2F2%29%2Aln%28x%29%2C+dx%29 ALIGN=MIDDLE  ALT=\"int%28x%5E%281%2F2%29%2Aln%28x%29%2C+dx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Let's use <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dv+=+x%5E%281%2F2%29dx ALIGN=MIDDLE  ALT=\"dv+=+x%5E%281%2F2%29dx\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v+=+%282%2F3%29x%5E%283%2F2%29 ALIGN=MIDDLE  ALT=\"v+=+%282%2F3%29x%5E%283%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > again. This time <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u+=+ln%28x%29 ALIGN=MIDDLE  ALT=\"u+=+ln%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=du+=+%281%2Fx%29dx ALIGN=MIDDLE  ALT=\"du+=+%281%2Fx%29dx\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28x%5E%281%2F2%29%2Aln%28x%29%2C+dx%29+=+ln%28x%29%2A%28%282%2F3%29x%5E%283%2F2%29%29+-+int%28%282%2F3%29x%5E%283%2F2%29%281%2Fx%29%2C+dx%29+=+ln%28x%29%2A%28%282%2F3%29x%5E%283%2F2%29%29+-+int%28%282%2F3%29x%5E%281%2F2%29%2C+dx%29+=+ln%28x%29%2A%28%282%2F3%29x%5E%283%2F2%29%29+-+%284%2F9%29x%5E%283%2F2%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "(Note how, after this second integration by parts, that we ended up with an \"easy\" integral: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28%282%2F3%29x%5E%281%2F2%29%2C+dx%29.%29%0D%0AWe+can+now+substitute+this+back+into+our+first+integration+by+parts%3A%0D%0A%7B%7B%7Bint%28%28ln%28x%29%29%5E2%2Asqrt%28x%29%2C+dx%29+=+%28%28ln%28x%29%29%5E2%29%2A%282%2F3%29x%5E%283%2F2%29+-+%284%2F3%29int%28%28x%5E%281%2F2%29%29%2A%28ln%28x%29%29%2C+dx%29+=+%28%28ln%28x%29%29%5E2%29%28%282%2F3%29x%5E%283%2F2%29+-+%284%2F3%29%28ln%28x%29%2A%282%2F3%29x%5E%283%2F2%29+-+%284%2F9%29x%5E%283%2F2%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "Simplifying we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28%28ln%28x%29%29%5E2%2Asqrt%28x%29%2C+dx%29+=+%282%2F3%29%28ln%28x%29%29%5E2%2Ax%5E%283%2F2%29+-+%288%2F9%29ln%28x%29%2Ax%5E%283%2F2%29+%2B+%2816%2F27%29x%5E%283%2F2%29+%2B+C ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And if you differentiate this you get, believe it or not,: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28ln%28x%29%29%5E2%2Asqrt%28x%29 ALIGN=MIDDLE  ALT=\"%28ln%28x%29%29%5E2%2Asqrt%28x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >! </SPAN>\n" );
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