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Math Help - Integration by Parts problem

  1. #1
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    Integration by Parts problem

    Hi guys, I have to teach a lesson on this tomorrow (trainee teacher, gulp!) and can't figure out this problem.

    Integrate:

    x^2sec^2xtanx.dx

    I can't seem to figure out if there's a way to manipulate this to make integration by parts easier. Any ideas? Would making it x^2sec.secxtanx help at all given that secxtanx integrates to sec x?

    Thanks in advance for any help.
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  2. #2
    Member integral's Avatar
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    Hint: Sec^2(x)=tan^2(x)+1
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  3. #3
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    Ouch. This - is a long one. You will need to use integration by parts twice - it is painless though - and a substitution in one of your parts:

    \int x^{2}sec^{2}(x)tan(x)dx

    u = x^{2}, du=2x

    \int dv = \int sec(x)sec(x)tan(x)dx

    Using substitution:

    w=sec(x), dw=sec(x)tan(x)dx

    v = \frac{sec^{2}(x)}{2}

    Setting up the by parts:

    \int x^{2}sec^{2}(x)tan(x)dx = \frac{x^{2}sec^{2}(x)}{2} -\int xsec^{2}(x)dx

    Taking integration by parts on the last integral:

    u = x, du = 1dx

     dv = sec^{2}(x), v = tan(x)

    Then (I'm only going to do the last part):

    \int xsec^{2}(x)dx = xtan(x) - \int tan(x)dx \Rightarrow

     \int xsec^{2}(x)dx = xtan(x) +ln |cos(x)|

    All together now:

    \int x^{2}sec^{2}(x)tan(x)dx = \frac{x^{2}sec^{2}(x)}{2} - xtan(x) - ln |cos(x)|
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