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May i know how to integrate [1 / (4 + x^2)^1/2] using trig?

  1. Jan 22, 2006 #1
    may i know how to integrate [1 / (4 + x^2)^1/2] using trigo substitution??

    pls help........
     
  2. jcsd
  3. Jan 22, 2006 #2

    TD

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    Well, you have a few trig identities, see which one is useful.
    Here, you could use the fact that [itex]\sec ^2 x = 1 + \tan ^2 x[/itex].

    So try the substitution [itex]x = 2\tan y \Leftrightarrow dx = \frac{2}{{\cos ^2 y}}dy[/itex]
     
  4. Jan 22, 2006 #3
    oh,i got it....can u pls tell me how do you know which frigo identities to choose when u you see the question in general??(not the question above)

    pls.....
     
  5. Jan 22, 2006 #4

    TD

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    For trigonometric substitutions, there are two fundamental identities which are often used:

    [tex]\begin{array}{l}
    \cos ^2 x + \sin ^2 x = 1 \Leftrightarrow \cos ^2 x = 1 - \sin ^2 x \\
    \sec ^2 x = 1 + \tan ^2 x \Leftrightarrow \tan ^2 x = \sec ^2 x - 1 \\
    \end{array}[/tex]

    The first one can be used for radical expressions of the form [itex]\sqrt {a^2 - x^2 } [/itex] where you then choose the substitution [itex]x = a\sin y[/itex].
    The second one can be used for two types: yours, which was of the form [itex]\sqrt {x^2 + a^2 } [/itex] (you then do [itex]x = a\tan y[/itex]) or those of the form [itex]\sqrt {x^2 - a^2 }[/itex] (then it's [itex]x = a\sec y[/itex]).
     
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