document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=211741>Question 211741</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi all, I was hoping to get some help with the following 2 intergrals.<BR>\n" );
document.write( "I have to evaluate each one using the given substitution in each.<BR>\n" );
document.write( "a) &#8747;( (x^2x) / ((3 + x^3)^2) )  Substitution, u = x^3<BR>\n" );
document.write( "b) &#8747; \( x^7 cos x^8dx)  Substitution  u = x^8<BR>\n" );
document.write( "Help with steps and notes on how to solve would be gret.<BR>\n" );
document.write( "Thanks, -Nick. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #159988 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=159988\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a) &#8747;( (x^2x) / ((3 + x^3)^2) )  Substitution, u = x^3<br><BR>\n" );
document.write( "I assume this is supposed to be:<BR>\n" );
document.write( "a) &#8747; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28+%28%28x%5E2%29dx%29+%2F+%28%283+%2B+x%5E3%29%5E2%29+%29 ALIGN=MIDDLE  ALT=\"%28+%28%28x%5E2%29dx%29+%2F+%28%283+%2B+x%5E3%29%5E2%29+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >  Substitution, u = x^3<BR>\n" );
document.write( "Actually a better substitution would be: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u+=+3+%2B+x%5E3 ALIGN=MIDDLE  ALT=\"u+=+3+%2B+x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > but what you were given will still work. We start by finding the derivative of both sides of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u+=+x%5E3 ALIGN=MIDDLE  ALT=\"u+=+x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28du%29%2F%28dx%29+=+3x%5E2 ALIGN=MIDDLE  ALT=\"%28du%29%2F%28dx%29+=+3x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Multiplying both sides of this by dx we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=du+=+3x%5E2%2Adx ALIGN=MIDDLE  ALT=\"du+=+3x%5E2%2Adx\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Looking at the original integral we can see the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Adx%29 ALIGN=MIDDLE  ALT=\"x%5E2%2Adx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > but not the 3. So we can \"move\" the 3 by multiplying both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F3 ALIGN=MIDDLE  ALT=\"1%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" > (or divide both sides by 3):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F3%29du+=+x%5E2%2Adx ALIGN=MIDDLE  ALT=\"%281%2F3%29du+=+x%5E2%2Adx\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we can substitute u for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E3 ALIGN=MIDDLE  ALT=\"x%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F3%29du ALIGN=MIDDLE  ALT=\"%281%2F3%29du\"   DISABLED_style= \"max-width:90%; height:auto;\" > for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2%2Adx ALIGN=MIDDLE  ALT=\"x%5E2%2Adx\"   DISABLED_style= \"max-width:90%; height:auto;\" > giving:<BR>\n" );
document.write( "a) &#8747; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28+%28%281%2F3%29du%29+%2F+%28%283+%2B+u%29%5E2%29+%29 ALIGN=MIDDLE  ALT=\"%28+%28%281%2F3%29du%29+%2F+%28%283+%2B+u%29%5E2%29+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This can be rewritten as:<BR>\n" );
document.write( "a) &#8747; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F3%29%28%283+%2B+u%29%5E-2%29du%29 ALIGN=MIDDLE  ALT=\"%281%2F3%29%28%283+%2B+u%29%5E-2%29du%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If the integral above is not obvious to you, then use a second substitution: v = 3+u. Either way we should get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2F3%29%281%2F%28-2+%2B+1%29%29%283+%2B+u%29%5E%28-2+%2B+1%29+%2B+C+=+%28%28-1%29%2F3%29%283+%2B+u%29%5E-1+%2B+C+=+%28-1%29%2F%283%283+%2B+u%29%29+%2B+C+=+%28-1%29%2F%289+%2B+3x%5E3%29+%2B+C ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "b) &#8747; \( x^7 cos x^8dx)  Substitution  u = x^8<BR>\n" );
document.write( "I assume the \"\\" is a typo and that the integral is:<BR>\n" );
document.write( "b) &#8747; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28x%5E7%2Acos%28x%5E8%29%2Adx%29 ALIGN=MIDDLE  ALT=\"%28x%5E7%2Acos%28x%5E8%29%2Adx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Again we start by finding the derivative of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u+=+x%5E8 ALIGN=MIDDLE  ALT=\"u+=+x%5E8\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28du%29%2F%28dx%29+=+x%5E7 ALIGN=MIDDLE  ALT=\"%28du%29%2F%28dx%29+=+x%5E7\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Multiplying both sides by dx:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=du+=+x%5E7%2Adx ALIGN=MIDDLE  ALT=\"du+=+x%5E7%2Adx\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Using the Commutative Property on the original integral we can see the right side of the above equation in the integral:<BR>\n" );
document.write( "&#8747; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cos%28x%5E8%29+%2A+x%5E7%2Adx%29 ALIGN=MIDDLE  ALT=\"%28cos%28x%5E8%29+%2A+x%5E7%2Adx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So we can now substitute u for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E8 ALIGN=MIDDLE  ALT=\"x%5E8\"   DISABLED_style= \"max-width:90%; height:auto;\" > and du for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E7%2Adx ALIGN=MIDDLE  ALT=\"x%5E7%2Adx\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "&#8747; <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28cos%28u%29+%2A+du%29 ALIGN=MIDDLE  ALT=\"%28cos%28u%29+%2A+du%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is a pretty simple integral to find:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28u%29+%2B+C+=+sin%28x%5E8%29+%2B+C ALIGN=MIDDLE  ALT=\"sin%28u%29+%2B+C+=+sin%28x%5E8%29+%2B+C\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );