document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208301>Question 1208301</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi can you help please:<BR>\n" );
document.write( "int (3 x^3) (sqrt(16-x^2)) dx<BR>\n" );
document.write( "Just not sure if my trig substitutions are working </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #846633 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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=846633\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Here is a page discussing various trig substitution rules<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://tutorial.math.lamar.edu/Classes/CalcII/TrigSubstitutions.aspx\">https://tutorial.math.lamar.edu/Classes/CalcII/TrigSubstitutions.aspx</a>\r<BR>\n" );
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document.write( "The rule we'll use specifically is <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28a%2Fb%29%2Asin%28theta%29 ALIGN=MIDDLE  ALT=\"x+=+%28a%2Fb%29%2Asin%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is in the blue box underneath example 4 on that web link I posted.<BR>\n" );
document.write( "Notice this rule corresponds to the template <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28a%5E2-b%5E2%2Ax%5E2%29 ALIGN=MIDDLE  ALT=\"sqrt%28a%5E2-b%5E2%2Ax%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "In this case a = 4 and b = 1.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28a%2Fb%29%2Asin%28theta%29 ALIGN=MIDDLE  ALT=\"x+=+%28a%2Fb%29%2Asin%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%284%2F1%29%2Asin%28theta%29 ALIGN=MIDDLE  ALT=\"x+=+%284%2F1%29%2Asin%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+4%2Asin%28theta%29 ALIGN=MIDDLE  ALT=\"x+=+4%2Asin%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Let's call this equation (1) to use later.\r<BR>\n" );
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document.write( "Then,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16-x%5E2+=+16+-+%284%2Asin%28theta%29%29%5E2 ALIGN=MIDDLE  ALT=\"16-x%5E2+=+16+-+%284%2Asin%28theta%29%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16-x%5E2+=+16+-+16%2Asin%5E2%28theta%29 ALIGN=MIDDLE  ALT=\"16-x%5E2+=+16+-+16%2Asin%5E2%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16-x%5E2+=+16%281+-+sin%5E2%28theta%29%29 ALIGN=MIDDLE  ALT=\"16-x%5E2+=+16%281+-+sin%5E2%28theta%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=16-x%5E2+=+16%2Acos%5E2%28theta%29 ALIGN=MIDDLE  ALT=\"16-x%5E2+=+16%2Acos%5E2%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2816-x%5E2%29+=+sqrt%28+16%2Acos%5E2%28theta%29+%29 ALIGN=MIDDLE  ALT=\"sqrt%2816-x%5E2%29+=+sqrt%28+16%2Acos%5E2%28theta%29+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2816-x%5E2%29+=+4%2Acos%28theta%29 ALIGN=MIDDLE  ALT=\"sqrt%2816-x%5E2%29+=+4%2Acos%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Let's call this equation (2) to use later.\r<BR>\n" );
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document.write( "Also<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+4%2Asin%28theta%29 ALIGN=MIDDLE  ALT=\"x+=+4%2Asin%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dx%2Fdtheta+=+4%2Acos%28theta%29 ALIGN=MIDDLE  ALT=\"dx%2Fdtheta+=+4%2Acos%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dx+=+4%2Acos%28theta%29%2Adtheta ALIGN=MIDDLE  ALT=\"dx+=+4%2Acos%28theta%29%2Adtheta\"   DISABLED_style= \"max-width:90%; height:auto;\" > Let's call this equation (3) to use later.<BR>\n" );
document.write( "Replace any mention of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dtheta ALIGN=MIDDLE  ALT=\"dtheta\"   DISABLED_style= \"max-width:90%; height:auto;\" > with <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d%2Atheta ALIGN=MIDDLE  ALT=\"d%2Atheta\"   DISABLED_style= \"max-width:90%; height:auto;\" >, but there won't be a dot between the d and theta. I couldn't get it to format properly.\r<BR>\n" );
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document.write( "Let's apply those items into the integral we want to evaluate.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28+3x%5E3%2Asqrt%2816-x%5E2%29%2C+dx+%29 ALIGN=MIDDLE  ALT=\"int%28+3x%5E3%2Asqrt%2816-x%5E2%29%2C+dx+%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%28+3%28highlight%28x%29%29%5E3%2Ahighlight%28sqrt%2816-x%5E2%29%29%2Ahighlight%28dx%29%2C+%22%22+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Each red box represents a thing we'll replace in the next line.<BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28+3%28highlight%284%2Asin%28theta%29%29%29%5E3%2Ahighlight%284%2Acos%28theta%29%29%2Ahighlight%284%2Acos%28theta%29%2Adtheta%29%2C+%22%22+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > Substitute in equations (1), (2), and (3).<BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%28+3%284%2Asin%28theta%29%29%5E3%2A4%2Acos%28theta%29%2A4%2Acos%28theta%29%2Adtheta%2C+%22%22+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3072%2Aint%28+sin%5E3%28theta%29%2Acos%5E2%28theta%29%2Adtheta%2C+%22%22+%29 ALIGN=MIDDLE  ALT=\"3072%2Aint%28+sin%5E3%28theta%29%2Acos%5E2%28theta%29%2Adtheta%2C+%22%22+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3072%2Aint%28+sin%28theta%29%2Asin%5E2%28theta%29%2Acos%5E2%28theta%29%2Adtheta%2C+%22%22+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3072%2Aint%28+sin%28theta%29%2A%281-cos%5E2%28theta%29%29%2Acos%5E2%28theta%29%2Adtheta%2C+%22%22+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3072%2Aint%28+%281-cos%5E2%28theta%29%29%2Acos%5E2%28theta%29%2Asin%28theta%29%2Adtheta%2C+%22%22+%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3072%2Aint%28+%281-u%5E2%29%2Au%5E2%2A%28-du%29%2C+%22%22+%29 ALIGN=MIDDLE  ALT=\"3072%2Aint%28+%281-u%5E2%29%2Au%5E2%2A%28-du%29%2C+%22%22+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Let u = cos(theta), which means du = -sin(theta)*dtheta and sin(theta)*dtheta = -du<BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3072%2Aint%28+%28u%5E4-u%5E2%29%2C+du+%29 ALIGN=MIDDLE  ALT=\"3072%2Aint%28+%28u%5E4-u%5E2%29%2C+du+%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3072%2A%28%28u%5E5%29%2F5+-+%28u%5E3%29%2F3%29%2BC ALIGN=MIDDLE  ALT=\"3072%2A%28%28u%5E5%29%2F5+-+%28u%5E3%29%2F3%29%2BC\"   DISABLED_style= \"max-width:90%; height:auto;\" > Don't forget about the plus C. I've seen many students make this error.<BR>\n" );
document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3072%28%28u%5E3%29%2F15%29%2A%283u%5E2+-+5%29%2BC ALIGN=MIDDLE  ALT=\"3072%28%28u%5E3%29%2F15%29%2A%283u%5E2+-+5%29%2BC\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "From here you'll need to replace each \"u\" with an expression of x.<BR>\n" );
document.write( "Recall that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u+=+cos%28theta%29 ALIGN=MIDDLE  ALT=\"u+=+cos%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%2816-x%5E2%29+=+4%2Acos%28theta%29 ALIGN=MIDDLE  ALT=\"sqrt%2816-x%5E2%29+=+4%2Acos%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Which would mean <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=cos%28theta%29+=+%281%2F4%29%2Asqrt%2816-x%5E2%29 ALIGN=MIDDLE  ALT=\"cos%28theta%29+=+%281%2F4%29%2Asqrt%2816-x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=u+=+%281%2F4%29%2Asqrt%2816-x%5E2%29 ALIGN=MIDDLE  ALT=\"u+=+%281%2F4%29%2Asqrt%2816-x%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "I'll let the student take over from here. \r<BR>\n" );
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document.write( "You should arrive at the final answer of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-1%2F5%29%2A%2816-x%5E2%29%5E%28%223%2F2%22%29%2A%283x%5E2%2B32%29%2BC ALIGN=MIDDLE  ALT=\"%28-1%2F5%29%2A%2816-x%5E2%29%5E%28%223%2F2%22%29%2A%283x%5E2%2B32%29%2BC\"   DISABLED_style= \"max-width:90%; height:auto;\" > or some equivalent variation of this. \r<BR>\n" );
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document.write( "You can use various online calculators (eg: WolframAlpha or GeoGebra) to confirm the answer.<BR>\n" );
document.write( "Here's what it looks like when using WolframAlpha<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.wolframalpha.com/input?i=int%28+3x%5E3*sqrt%2816-x%5E2%29%2C+dx+%29\">https://www.wolframalpha.com/input?i=int%28+3x%5E3*sqrt%2816-x%5E2%29%2C+dx+%29</a><BR>\n" );
document.write( "The input typed in was <font color=red>int( 3x^3*sqrt(16-x^2), dx )</font> or you could do <font color=red>integral( 3x^3*sqrt(16-x^2), dx )</font>\r<BR>\n" );
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document.write( "More practice found here<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.algebra.com/algebra/homework/Exponential-and-logarithmic-functions/Exponential-and-logarithmic-functions.faq.question.1208255.html\">https://www.algebra.com/algebra/homework/Exponential-and-logarithmic-functions/Exponential-and-logarithmic-functions.faq.question.1208255.html</a><BR>\n" );
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