document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1191333>Question 1191333</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Evalute integral sin^2 (x) * cos^2 (x) dx from bounds 0 to pi/3 <BR>\n" );
document.write( "by using : cos^2 (x) = 1 - sin^2 (x) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #823165 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=823165\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Before we get to the calculus, here are some useful power reduction trig identities.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \cos^2(\text{x}) = \frac{1+\cos(2\text{x})}{2} = 0.5(1+\cos(2\text{x}))\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \sin^2(\text{x}) = \frac{1-\cos(2\text{x})}{2} = 0.5(1-\cos(2\text{x}))\">\r<BR>\n" );
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document.write( "If we were to square both sides of the second identity, then we have:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \sin^2(\text{x}) = \frac{1-\cos(2\text{x})}{2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \left(\sin^2(\text{x})\right)^2 = \left(\frac{1-\cos(2\text{x})}{2}\right)^2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \sin^4(\text{x}) = \frac{\left(1-\cos(2\text{x})\right)^2}{4}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \sin^4(\text{x}) = \frac{1-2\cos(2\text{x})+\cos^2(2\text{x})}{4}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \sin^4(\text{x}) = \frac{1-2\cos(2\text{x})+0.5(1+\cos(4\text{x}))}{4}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \sin^4(\text{x}) = \frac{1-2\cos(2\text{x})+0.5+0.5\cos(4\text{x})}{4}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \sin^4(\text{x}) = \frac{1.5-2\cos(2\text{x})+0.5\cos(4\text{x})}{4}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \sin^4(\text{x}) = \frac{0.5(3-4\cos(2\text{x})+\cos(4\text{x}))}{4}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \sin^4(\text{x}) = \frac{3-4\cos(2\text{x})+\cos(4\text{x})}{8}\">\r<BR>\n" );
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document.write( "which will be useful later.\r<BR>\n" );
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document.write( "Now to rewrite the integral using those trig identities and cos^2 (x) = 1 - sin^2 (x)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \int_{0}^{\pi/3}\sin^2(\text{x})\cos^2(\text{x})d\text{x}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \int_{0}^{\pi/3}\sin^2(\text{x})\left(1-\sin^2(\text{x})\right)d\text{x}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \int_{0}^{\pi/3}\left(\sin^2(\text{x})-\sin^4(\text{x})\right)d\text{x}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \int_{0}^{\pi/3}\sin^2(\text{x})d\text{x}-\int_{0}^{\pi/3}\sin^4(\text{x})d\text{x}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\large \int_{0}^{\pi/3}\frac{1-\cos(2\text{x})}{2}d\text{x}-\int_{0}^{\pi/3}\frac{3-4\cos(2\text{x})+\cos(4\text{x})}{8}d\text{x}\">\r<BR>\n" );
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document.write( "I'll let you finish up from here. You'll need to use u-substitution at some parts.<BR>\n" );
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