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Math Help - Indefinite integration

  1. #1
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    Indefinite integration

    INTEGRATE (x sec x)^2 dx
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    MHF Contributor FernandoRevilla's Avatar
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    Re: Indefinite integration

    Express \int (x\sec x)^2\;dx=\int x^2\sec^2 x\;dx and use integration by parts with u=x^2,\;dv=\sec^2 x\;dx.
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    Re: Indefinite integration

    Quote Originally Posted by kyou View Post
    INTEGRATE (x sec x)^2 dx
    \displaystyle \begin{align*} \int{\left(x \sec{x}\right)^2\,dx} &= \int{x^2\sec^2{x}\,dx} \end{align*}

    Use integration by parts with \displaystyle \begin{align*} u = x^2 \implies du = 2x\,dx \end{align*} and \displaystyle \begin{align*} dv = \sec^2{x}\,dx \implies v = \tan{x} \end{align*} and the integral becomes

    \displaystyle \begin{align*} \int{x^2\sec^2{x}\,dx} &= x^2\tan{x} - \int{2x\tan{x}\,dx} \\ &= x^2\tan{x} - 2\int{x\tan{x}\,dx} \end{align*}

    Now use integration by parts again with \displaystyle \begin{align*} u = x \implies du = dx \end{align*} and \displaystyle \begin{align*} dv = \tan{x}\,dx \implies v = \ln{|\sec{x}|} \end{align*} and the integral becomes

    \displaystyle \begin{align*} x^2\tan{x} - 2\int{x\tan{x}\,dx} &=  x^2\tan{x} - 2\left(x\ln{|\sec{x}|} - \int{\ln{|\sec{x}|}\,dx}\right) \\ &= x^2\tan{x} - 2x\ln{|\sec{x}|} - \int{\ln{|\sec{x}|}\,dx} \end{align*}

    This does not have a closed form solution.
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    Re: Indefinite integration

    That leaves me requiring integral of xtanx (?)
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    Re: Indefinite integration

    Quote Originally Posted by biffboy View Post
    That leaves me requiring integral of xtanx (?)
    Did you even read my post? I showed you that the integral does not have a closed form solution...
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    Re: Indefinite integration

    My post was before yours :-)
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    Re: Indefinite integration

    No it wasn't. Apologies.
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    Re: Indefinite integration

    Quote Originally Posted by biffboy View Post
    My post was before yours :-)
    Was it really?

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    Re: Indefinite integration

    Well, even if the indefinite integral does not have a closed form evaluation , it does have an infinite series representation , cause we can express tan x as an infinite series in terms of x and then we can obviously multiply the series by x and integrate term by term , owing to the convergence of the series , thus we get the indefinite integral
    of xtanx and hence that of (xsecx)^2
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  10. #10
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    Re: Indefinite integration

    My reply was to the post from Spain.
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