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Math Help - Alternative methods to solving the integral of [tan(x)]^6 * sec(x )

  1. #1
    Newbie ElBartoV's Avatar
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    Alternative methods to solving the integral of [tan(x)]^6 * sec(x )

    I was just curious to see if there was alternative methods to solving

    \displaystyle\int \tan^6(x)\sec(x)\,dx

    asides from using integrations by parts since it tends to yield a very lengthy solution. Although, I believe the reduction formula for sec(x) could probably be applied in this case instead of integration by parts

    \displaystyle\int\sec^n(x)\,dx = -\frac{1}{n-1}\cot^{n-1}(x) - \cot^{n-2}(x)

    but, still is there any other methods asides from these two that could be used?



    (P.S. sorry if the format is not completely correct, this is the first time i have used LaTex)
    Last edited by Chris L T521; August 21st 2010 at 10:08 PM. Reason: Fixed Latex (@ElBartoV: To see my corrections, click "edit post" button)
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by ElBartoV View Post
    I was just curious to see if there was alternative methods to solving

    \displaystyle\int \tan^6(x)\sec(x)\,dx

    asides from using integrations by parts since it tends to yield a very lengthy solution. Although, I believe the reduction formula for sec(x) could probably be applied in this case instead of integration by parts

    \displaystyle\int\sec^n(x)\,dx = -\frac{1}{n-1}\cot^{n-1}(x) - \cot^{n-2}(x)

    but, still is there any other methods asides from these two that could be used?



    (P.S. sorry if the format is not completely correct, this is the first time i have used LaTex)
    I tried to fix the tex for you...did I interpret everything correctly? I don't think I did... XD
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  3. #3
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    \tan^2{x} = \sec^2{x} - 1.

    So \tan^6{x} = (\tan^2{x})^3

     = (\sec^2{x} - 1)^3

     = \sec^6{x} - 3\sec^4{x} + 3\sec^2{x} - 1.


    Therefore \int{\tan^6{x}\sec{x}\,dx} = \int{(\sec^6{x} - 3\sec^4{x} + 3\sec^2{x} - 1)\sec{x}\,dx}

     = \int{\sec^7{x} - 3\sec^5{x} + 3\sec^3{x} - \sec{x}\,dx}.


    Now you can integrate term by term using the reduction method you posted above.
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  4. #4
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    Quote Originally Posted by ElBartoV View Post
    I was just curious to see if there was alternative methods to solving

    \displaystyle\int \tan^6(x)\sec(x)\,dx

    asides from using integrations by parts since it tends to yield a very lengthy solution. Although, I believe the reduction formula for sec(x) could probably be applied in this case instead of integration by parts

    \displaystyle\int\sec^n(x)\,dx = -\frac{1}{n-1}\cot^{n-1}(x) - \cot^{n-2}(x)

    but, still is there any other methods asides from these two that could be used?



    (P.S. sorry if the format is not completely correct, this is the first time i have used LaTex)
    Although still long, you could write the integral as

    \displaystyle \int \frac{\sin^6x}{\cos^7x}dx = \int \frac{\sin^6x \cos x}{\cos^8x}dx

    then let u = \sin x giving

    \displaystyle \int \frac{u^6}{(1-u^2)^4}du = \int \frac{u^6}{(1-u)^4(1+u)^4}du

    then partial fractions.
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  5. #5
    Newbie ElBartoV's Avatar
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    The method of partial fractions looks to be a little less lengthy then using the reduction formula or at the very least itís easier to keep track of all the terms with only the variable u (a whole bunch of factors of sec(x) can get confusing sometimes). Also thank you for the tex fix, Iím still getting use to the LaTex system.
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