document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=203271>Question 203271</A>: <I><SPAN STYLE=\"word-break: break-all;\"> i want the integral of dx/(2+sinx) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #153370 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=153370\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is a tricky one. It involves a substitution which \"converts\" rational<BR>\n" );
document.write( "functions of sin and/or cos into \"plain\" rational functions which may be<BR>\n" );
document.write( "more easily integrated. The substituion is based on a right triangle with<BR>\n" );
document.write( "legs of 2z and (1-z^2) and a hypotenuse of (1+z^2). And let the angle<BR>\n" );
document.write( "between the hypoentuse and the (1-z^2) leg be called x. In this triangle:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sin%28x%29+=+%282z%29%2F%281%2Bz%5E2%29 ALIGN=MIDDLE  ALT=\"sin%28x%29+=+%282z%29%2F%281%2Bz%5E2%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%28x%29+=+%281-z%5E2%29%2F%281%2Bz%5E2%29 ALIGN=MIDDLE  ALT=\"cos%28x%29+=+%281-z%5E2%29%2F%281%2Bz%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And, thru the use of 1/2 angle trig identities:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2+arctan%28z%29 ALIGN=MIDDLE  ALT=\"x+=+2+arctan%28z%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or z = tan((1/2)x)<BR>\n" );
document.write( "which leads to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dx+=+%282%2F%281%2Bz%5E2%29%29dz ALIGN=MIDDLE  ALT=\"dx+=+%282%2F%281%2Bz%5E2%29%29dz\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now we can substitute in for sin(u) and, if we had one, cos(u), and for du:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%281%2F%282+%2B+sin%28x%29%29%2C+dx%29+=+int%281%2F%282+%2B+%282z%29%2F%281%2Bz%5E2%29%29%2A%282%2F%281%2Bz%5E2%29%29%2C+dz%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Multiplying the numerator and denominaor of the fraction by (1+z^2) we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%282%2F%28%281%2Bz%5E2%29%2A2+%2B+2z%29%2C+dz%29+=+int%282%2F%282%2B2z%5E2%2B2z%29%2C+dz%29+=+int%282%2F%282%28z%5E2+%2B+z%0D%0A%2B+1%29%29%2C+dz%29+=+int%281%2F%28z%5E2+%2B+z+%2B+1%29%2C+dz%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "And, if we complete the square in the denominator we can have an integral<BR>\n" );
document.write( "of the du/(a^2 + u^2) variety (an arctan):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%281%2F%28z%5E2+%2B+z+%2B+1%29%2C+dz%29+=+int%281%2F%28%283%2F4%29+%2B+%28z%5E2+%2B+z+%2B+1%2F4%29%29%2C+dz%29+=%0D%0Aint%281%2F%28%28sqrt%283%29%2F2%29%5E2+%2B+%28z+%2B+%281%2F2%29%5E2%29%29%2C+dz%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This fits the pattern of the intergation formula: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%281%2F%28a%5E2+%2B+u%5E2%29%2C%0D%0Adu%29+=+%281%2Fa%29+arctan%28%28u%2Fa%29+%2B+c%29 ALIGN=MIDDLE  ALT=\"int%281%2F%28a%5E2+%2B+u%5E2%29%2C%0D%0Adu%29+=+%281%2Fa%29+arctan%28%28u%2Fa%29+%2B+c%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=int%281%2F%28z%5E2+%2B+z+%2B+1%29%2C+dz%29+=+%282%2Fsqrt%283%29%29arctan%282%28z+%2B+1%2F2%29%2Fsqrt%283%29%29+%2B+C%0D%0A=+%282%2Fsqrt%283%29%29arctan%28%282z+%2B+1%29%2Fsqrt%283%29%29+%2B+C+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Substituting back in for z we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%2Fsqrt%283%29%29arctan%282%28tan%28%281%2F2%29x%29+%2B+1%29%2Fsqrt%283%29%29+%2B+C+ ALIGN=MIDDLE  ALT=\"%282%2Fsqrt%283%29%29arctan%282%28tan%28%281%2F2%29x%29+%2B+1%29%2Fsqrt%283%29%29+%2B+C+\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "of, if you like rationalized denominators:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282sqrt%283%29%2F3%29arctan%282sqrt%283%29%28tan%28%281%2F2%29x%29+%2B+sqrt%283%29%29%2F3%29+%2B+C+ ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "the derivative of which, believe it or not, is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F%282%2Bsin%28x%29%29 ALIGN=MIDDLE  ALT=\"1%2F%282%2Bsin%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >! </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );