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Easy calc problem

  1. Mar 30, 2005 #1
    Hey, i know this is an easy problem, but i don't have any of my older calc books with the identities, so, just hoping for a quick answer

    integral(sec^3 X)

    thanks... i just need that for a more difficult problem... so i don't want to spend a lot on this simple integral, thanks.
     
  2. jcsd
  3. Mar 30, 2005 #2
  4. Mar 30, 2005 #3
    let [tex] I = \int sec^3xdx [/tex]

    and let

    [tex] u = secx \ \ \Rightarrow \ \ du = secxtanxdx [/tex]

    [tex] dv = sec^2x \ \ \Rightarrow \ \ v = tanx [/tex]

    using integration by parts we have

    [tex] I = uv - \int vdu [/tex]

    [tex] \Rightarrow \ \ I = secxtanx - \int secxtan^2xdx [/tex]

    using the trigonometric identity tan^2 = sec^2 -1 the integral on the right side becomes
    [tex] I = secxtanx - \int[secx(sec^2x - 1)]dx [/tex]

    [tex] \Rightarrow I = secxtanx - \int sec^3xdx + \int secxdx [/tex]

    [tex] \Rightarrow I = secxtanx - I + ln|tanx + secx| [/tex]

    [tex] \Rightarrow 2I = secxtanx + ln|tanx + secx| [/tex]

    [tex] \Rightarrow I = \frac{1}{2}[ \ secxtanx + ln|tanx + secx| \ ] + C[/tex]
     
    Last edited: Mar 30, 2005
  5. Mar 30, 2005 #4

    tony873004

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    Science Advisor
    Gold Member

  6. Mar 30, 2005 #5
    Derivatives are easy for all products, quotients, powers, compositions, and sums of elementary functions... antiderivatives are much more difficult.
     
  7. Mar 30, 2005 #6

    xanthym

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    Science Advisor

    YES ... Integrals, Derivatives, Graphing, and Matrices ... GO TO:
    -----> http://www.calc101.com/
    Select from list upper right corner home page. Remember to enter functions with SQUARE BRACKETS "[]", like "sin[x]", and group with PARENTHESES "()", like:
    "(x + 3*exp[5*x])*(x*sin[x])^3"



    ~~
     
    Last edited: Mar 30, 2005
  8. Mar 30, 2005 #7
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