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Integral Of

  1. Dec 18, 2004 #1
    whats the integral of tan^2(u)(sec(u))du?

    i was trying to integrate
    (x^2)/sqrt(x^2+1)dx, and came into that. it turns out pretty messy though, is there a clean way to do it?
     
  2. jcsd
  3. Dec 18, 2004 #2

    shmoe

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    Hi, I'm usually inclined to convert an integral like that to one with only sin's and cos's:

    [tex]\int\frac{\sin(u)^2}{\cos(u)^3}du[/tex]

    Integration by parts will work on this if you break it up properly.
     
  4. Dec 18, 2004 #3

    dextercioby

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    [tex]\int \frac{x^{2}}{\sqrt{x^{2}+1}} dx =...? [/tex]
    Make the natural substitution:[itex] x\rightarrow \sinh y [/itex]

    U'll be gettin' [tex] \int \sinh^{2}y dy [/tex] (1)
    Consider the "sister integral" [tex] \int \cosh^{2}y dy [/tex] (2)

    Consider the two expressions obtained by:(2)+(1);(2)-(1).The two new integrals will be trivials since u can use the 2 formulae from hyperbolic trigonometry:
    [tex] \cosh^{2}y-\sinh^{2}y=1;\cosh^{2}y+\sinh^{2}y= \cosh{2y} [/tex]

    In the end u can extract this integral [tex] \int \sinh^{2}y dy [/tex] easily and then in the final result u'll have to make the substitution back
    [tex] y\rightarrow \arg\sinh x [/tex]

    Daniel.
     
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