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    hi i'm really struggling to understand part of the solution to the following question:

    Integrate x^2(2sec^2x)tanx

    so i've got u = x^2 then du/dx = 2x
    and then dv/dx = 2secxsecxtanx but i don't really know how to integrate this to find v.

    the solution says that v is sec^2x but i'm really struggling to understand how that works.

    any help would be really appreciated! thanks.
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    (Original post by yellow.ammie)
    hi i'm really struggling to understand part of the solution to the following question:

    Integrate x^2(2sec^2x)tanx

    so i've got u = x^2 then du/dx = 2x
    and then dv/dx = 2secxsecxtanx but i don't really know how to integrate this to find v.

    the solution says that v is sec^2x but i'm really struggling to understand how that works.

    any help would be really appreciated! thanks.
    The intergral of sec^nxtanx with respect to x is 1/n(sec^nx) + c
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    (Original post by yellow.ammie)
    hi i'm really struggling to understand part of the solution to the following question:

    Integrate x^2(2sec^2x)tanx

    so i've got u = x^2 then du/dx = 2x
    and then dv/dx = 2secxsecxtanx but i don't really know how to integrate this to find v.

    the solution says that v is sec^2x but i'm really struggling to understand how that works.

    any help would be really appreciated! thanks.
    If you re-expressed \displaystyle 2\mathrm{sec}^2(x)\mathrm{tan}(x  ) = 2\frac{\mathrm{sin}(x)}{\mathrm{  cos}^3(x)}, then you could subsequently use a substitution to integrate it, I think?

    Substitution:
    Spoiler:
    Show
    t = \mathrm{cos}(x)


    EDIT: Or use the above poster's formula. :o:
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    (Original post by Clarity Incognito)
    The intergral of sec^nxtanx with respect to x is 1/n(sec^nx) + c

    is that a general rule then? my friend tried to explain it to me using the reverse of the chain rule but i don't quite get it...
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    (Original post by james.h)
    If you re-expressed \displaystyle 2\mathrm{sec}^2(x)\mathrm{tan}(x  ) = 2\frac{\mathrm{sin}(x)}{\mathrm{  cos}^3(x)}, then you could subsequently use a substitution to integrate it, I think?

    Substitution:
    Spoiler:
    Show
    t = \mathrm{cos}(x)


    EDIT: Or use the above poster's formula. :o:
    I wouldn't go down this route if I were you, you will get very muddled and the algebraic manipulation is not worth it. Use the result I've given above. It's derived from the fact that the differential of secx is secxtanx, otherwise you can use tanx = u as a substitution and integrate from there.
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    (Original post by yellow.ammie)
    is that a general rule then? my friend tried to explain it to me using the reverse of the chain rule but i don't quite get it...
    Yeah, it's using the result that d(secx)/dx is secxtanx. This you can get from using the quotient rule and differentiating 1/cosx.

    Now d/dx of sec^nx = nsec^n-1x multipled by (sec^nxtanx)

    which is equal to nsec^nxtanx.

    Applying this in reverse, you get the general rule.
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    (Original post by yellow.ammie)
    is that a general rule then? my friend tried to explain it to me using the reverse of the chain rule but i don't quite get it...
    I would note that it is a case of having a function next to the derivative of the function. i.e. a formula of the form f(x)f'(x). You get that kind of formula after differentiating ((f(x))^2. You should remember that you can differentiate (f(x))^2 to get 2f'(x)f(x) which is a formula in the form that you have (give or take a constant out the front). As differentiation is the 'opposite' of integration this kind of reasoning means that given a formula of the form f(x)f'(x), you can have a good guess at what the integral of the formula will be.
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    ok thank you for all the help!
 
 
 
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