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Integral of dx/ (1+cos ^2(x))

  1. May 12, 2005 #1
    for the life of me i cant seem to understand how to the the intergral of dx/ (1+cos ^2(x))????????
     
  2. jcsd
  3. May 12, 2005 #2

    dextercioby

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    Use a substitution

    [tex] \tan\frac{x}{2}=t [/tex]

    and some trigonometry.

    Daniel.
     
  4. May 12, 2005 #3

    arildno

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    Daniel's approach probably works equally well; here's another approach:
    [tex]\frac{1}{1+\cos^{2}x}=\frac{1}{\cos^{2}x}\frac{1}{1+\frac{1}{\cos^{2}x}}=(\frac{d}{dx}tan(x))\frac{1}{2+\tan^{2}x}[/tex]
    Thus, setting [tex]u=tan(x)[/tex], we have [tex]\frac{du}{dx}dx=du[/tex], that is:
    [tex]\int\frac{dx}{1+\cos^{2}x}=\int\frac{du}{2+u^{2}}[/tex]
     
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