document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1208255>Question 1208255</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hi, can you help with this question:<BR>\n" );
document.write( "Use x=alphasintheta for sqrt(alpha^2-x^2), x= alphatantheta for sqrt(alpha^2+x^2) and x= alphasectheta for sqrt(x^2-alpha^2), to find int (x/(sqrt(1-x^2)^3) dx </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #846560 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=846560\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "sqrt(1-x^2) is of the form sqrt(alpha^2-x^2) where alpha = 1.<BR>\n" );
document.write( "We use the trig substitution x = sin(theta) which leads to dx = cos(theta)*dtheta\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%281+-+x%5E2%29+=+sqrt%28+1+-+sin%5E2%28theta%29+%29 ALIGN=MIDDLE  ALT=\"sqrt%281+-+x%5E2%29+=+sqrt%28+1+-+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=sqrt%28+cos%5E2%28theta%29+%29 ALIGN=MIDDLE  ALT=\"sqrt%28+cos%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=cos%28theta%29 ALIGN=MIDDLE  ALT=\"cos%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "In short,<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%281-x%5E2%29+=+cos%28theta%29 ALIGN=MIDDLE  ALT=\"sqrt%281-x%5E2%29+=+cos%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > based on the set up <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+sin%28theta%29 ALIGN=MIDDLE  ALT=\"x+=+sin%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Apply the reciprocal to both sides to determine that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%28sqrt%281-x%5E2%29%29+=+1%2F%28cos%28theta%29%29+=+sec%28theta%29 ALIGN=MIDDLE  ALT=\"1%2F%28sqrt%281-x%5E2%29%29+=+1%2F%28cos%28theta%29%29+=+sec%28theta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "A drawing of a right triangle might be helpful. <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%0D%0Adrawing%28400%2C400%2C-1%2C4%2C-1%2C5%2C%0D%0Aline%280%2C0%2C3%2C0%29%2Cline%283%2C0%2C3%2C4%29%2Cline%283%2C4%2C0%2C0%29%2Cline%282.72%2C0%2C2.72%2C0.28%29%2Cline%282.72%2C0.28%2C3%2C0.28%29%2C%0D%0Alocate%280.74%2C0.54%2Ctheta%29%2Clocate%283.26%2C2.5%2B0.5%2C%22opposite%22%29%2Clocate%283.26%2C2.5%2C%22x%22%29%2Clocate%281.02-0.7%2C2.86-0.3%2C%22hypotenuse=1%22%29%2Clocate%281.44-0.8%2C-0.32%2Cadjacent=sqrt%281-x%5E2%29%29%0D%0A%29%0D%0A ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "sine = opposite/hypotenuse<BR>\n" );
document.write( "cosine = adjacent/hypotenuse<BR>\n" );
document.write( "secant = hypotenuse/adjacent\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=int%28x%2F%28%28sqrt%281-x%5E2%29%29%5E3%29%2Cdx%29+=+int%28%28sin%28theta%29%2Acos%28theta%29matrix%281%2C2%2Cd%2Ctheta%29%29%2F%28%28cos%28theta%29%29%5E3%29%2C%22%22%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%28%28sin%28theta%29matrix%281%2C2%2Cd%2Ctheta%29%29%2F%28%28cos%28theta%29%29%5E2%29%2C%22%22%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%28%281%2Fcos%28theta%29%29%2A%28sin%28theta%29%2Fcos%28theta%29%29matrix%281%2C2%2Cd%2Ctheta%29%2C%22%22%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%28sec%28theta%29%2Atan%28theta%29matrix%281%2C2%2Cd%2Ctheta%29%2C%22%22%29 ALIGN=MIDDLE  ALT=\"int%28sec%28theta%29%2Atan%28theta%29matrix%281%2C2%2Cd%2Ctheta%29%2C%22%22%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=sec%28theta%29%2BC ALIGN=MIDDLE  ALT=\"sec%28theta%29%2BC\"   DISABLED_style= \"max-width:90%; height:auto;\" > Don't forget the plus C.\r<BR>\n" );
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document.write( "= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%28+sqrt%281-x%5E2%29+%29+%2B+C ALIGN=MIDDLE  ALT=\"1%2F%28+sqrt%281-x%5E2%29+%29+%2B+C\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "We therefore conclude<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%281%2C3%2Cint%28x%2F%28%28sqrt%281-x%5E2%29%29%5E3%29%2Cdx%29%2C%22=%22%2C1%2F%28+sqrt%281-x%5E2%29+%29+%2B+C%29+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "You can use an online tool such as WolframAlpha to verify.<BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.wolframalpha.com/input?i=int%28x%2F%28%28sqrt%281-x%5E2%29%29%5E3%29%29\">https://www.wolframalpha.com/input?i=int%28x%2F%28%28sqrt%281-x%5E2%29%29%5E3%29%29</a><BR>\n" );
document.write( "The CAS mode in GeoGebra is another tool you could use.\r<BR>\n" );
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document.write( "Another way to verify is to apply the derivative to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%28+sqrt%281-x%5E2%29+%29+%2B+C ALIGN=MIDDLE  ALT=\"1%2F%28+sqrt%281-x%5E2%29+%29+%2B+C\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and you should get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%2F%28%28sqrt%281-x%5E2%29%29%5E3%29 ALIGN=MIDDLE  ALT=\"x%2F%28%28sqrt%281-x%5E2%29%29%5E3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "I'll leave this for the student to do.\r<BR>\n" );
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document.write( "More practice is found here<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><BR>\n" );
document.write( "<a  rel=nofollow HREF=\"https://www.algebra.com/algebra/homework/Exponential-and-logarithmic-functions/Exponential-and-logarithmic-functions.faq.question.1208301.html\">https://www.algebra.com/algebra/homework/Exponential-and-logarithmic-functions/Exponential-and-logarithmic-functions.faq.question.1208301.html</a>\r<BR>\n" );
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document.write( "Side note: There shouldn't be a space between the d and theta in the notation <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=matrix%281%2C2%2Cd%2Ctheta%29 ALIGN=MIDDLE  ALT=\"matrix%281%2C2%2Cd%2Ctheta%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > but I couldn't get it to render without that space.<BR>\n" );
document.write( "If the space wasn't there, then it would mistakenly render <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dtheta ALIGN=MIDDLE  ALT=\"dtheta\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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