document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=697886>Question 697886</A>: <I><SPAN STYLE=\"word-break: break-all;\"> integral of x cos^-1 x dx<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #430386 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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/cgi-bin/show-face.mpl?user=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=430386\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \int\,{x\,\arccos(x)\,dx}\">\r<BR>\n" );
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document.write( "Use Integration by Parts:\r<BR>\n" );
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document.write( "Let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ u\ =\ \arccos(x)\">\r<BR>\n" );
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document.write( "Then from any standard table of derivatives: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \frac{du}{dx}\ =\ -\frac{1}{\sqrt{1\ -\ x^2}}\">\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ du\ =\ -\frac{1}{\sqrt{1\ -\ x^2}}\,dx\">\r<BR>\n" );
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document.write( "And <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ dv\ =\ x\,dx\">\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ v\ =\ \int\,dv\ =\ \int\,x\,dx\ =\ \frac{x^2}{2}\ +\ C\">\r<BR>\n" );
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document.write( "Then substituting into the Integration by Parts formula:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \int\,{x\,\arccos(x)\,dx}\ =\ \frac{1}{2}x^2\,\arccos(x)\ +\ \frac{1}{2}\int\,\frac{x^2}{\sqrt{1\,-\,x^2}}\,dx\">\r<BR>\n" );
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document.write( "To calculate the remaining integral, let <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ x\ =\ \sin\varphi\">\r<BR>\n" );
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document.write( "Then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \frac{dx}{d\varphi}\ =\ \cos\varphi\">\r<BR>\n" );
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document.write( "Then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ dx\ =\ \cos\varphi\,d\varphi\">\r<BR>\n" );
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document.write( "And then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \int\,\frac{x^2}{\sqrt{1\,-\,x^2}}\,dx\ =\ \int\,\frac{\sin^2\varphi\,\cos\varphi}{\sqrt{1\,-\,\sin^2\varphi}}\,d\varphi\">\r<BR>\n" );
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document.write( "But since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \sqrt{1\ -\ \sin^2\theta}\ =\ \cos\theta\">,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \int\,\frac{x^2}{\sqrt{1\,-\,x^2}}\,dx\ =\ \int\,\sin^2\varphi\,d\varphi\">\r<BR>\n" );
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document.write( "Rewrite:  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \sin^2\varphi\ =\ \frac{1}{2}\left(1\ -\ \cos{2\varphi}\right)\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \int\,\sin^2\varphi\,d\varphi\ =\ \frac{1}{2}\int\,\left(1\ -\ \cos{2\varphi}\right)\,d\varphi\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ =\ \frac{1}{2}\int\,d\varphi\ -\ \frac{1}{2}\int\,\cos{2\varphi}\,d\varphi\ =\ \">\r<BR>\n" );
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document.write( "Then since <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ x\ =\ sin\varphi\">, <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \varphi\ =\ \arcsin(x)\"> and <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \cos\varphi\ =\ \sqrt{1\ -\ \sin^2\varphi}\ =\ \sqrt{1\ -\ x^2}\">, so:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \frac{1}{2}\int\,d\varphi\ -\ \frac{1}{2}\int\,2\sin\varphi\cos\varphi\,d\varphi\ =\ \frac{1}{2}\arcsin(x)\ -\ \frac{1}{2}x\sqrt{1\,-\,x^2}\ +\ C\">\r<BR>\n" );
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document.write( "Finally, putting the pieces back together:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \int\,{x\,\arccos(x)\,dx}\ =\ \frac{1}{2}x^2\arccos(x)\ +\ \frac{1}{4}\arcsin(x)\ -\ \frac{1}{4}x\sqrt{1\,-\,x^2}\ +\ C\">\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi}\ +\ 1\ =\ 0\"><BR>\n" );
document.write( "<font face=\"Math1\" size=\"+2\">Egw to Beta kai to Sigma</font><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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