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Math Help - Integration

  1. #1
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    Integration

    Integral e^xcos(2x)dx
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by gracy View Post
    Integral e^xcos(2x)dx
    If you had done some theory of complex numbers this would be trivial, but
    as we must assume otherwise you can do this using integration by parts.

    I = Integral e^xcos(2x)dx

    put dv=e^x and u=cos(2x), the the integration by parts rule gives:

    I = e^x cos(2x) + 2 integral e^x sin(2x) dx

    Now we do this again with the integral on the right, with dv=e^x and u=sin(2x),
    to get:

    I = e^x cos(2x) + 2 [e^x sin(2x) - 2 integral e^x cos(2x) dx]

    or:

    I = e^x cos(2x) + 2 e^x sin(2x) - 4 I,

    so:

    5I = e^x cos(2x) + 2 e^x sin(2x)

    or:

    I = [e^x cos(2x) + 2 e^x sin(2x)]/5

    and finally add in a constant of integration:

    I = [e^x cos(2x) + 2 e^x sin(2x)]/5 + C

    RonL
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  3. #3
    MHF Contributor
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    Quote Originally Posted by gracy View Post
    Integral e^xcos(2x)dx
    Integral e^xcos(2x)dx

    Integrand is a product of two unrelated functions, so try to integrate by parts:
    INT[u]dv = uv -INT[v]du ---------------***



    INT.[(e^x)cos(2x)]dx -------------------------(0)
    = INT.[cos(2x)][e^x dx] -----(i)

    Let u = cos(2x) --------------And, dv = e^x dx
    So, du = -2sin(2x) dx --------------v = e^x

    = cos(2x) *e^x -INT.[e^x][-2sin(2x) dx]
    = (e^x)cos(2x) +(2)INT.[sin(2x)][e^x dx] -------(ii)

    The integral part of (ii) is by parts again.
    Let r = sin(2x) ------------And, ds = e^x dx
    So, dr = 2cos(2x) dx ------------s = e^x

    = (e^x)cos(2x) +2{sin(2x) *ex -INT.[e^x][2cos(2x) dx]}
    = (e^x)cos(2x) + 2(e^x)sin(2x) -(4)INT.[(e^x)cos(2x)]dx ----------(iii)

    The integral part of (iii) is (-4) times the original integral (0), so collect like terms,

    INT.[(e^x)cos(2x)]dx +(4)INT.[(e^x)cos(2x)]dx = (e^x)cos(2x) +2(e^x)sin(2x)

    (5)INT.[(e^x)cos(2x)]dx = (e^x)cos(2x) +2(e^x)sin(2x)

    Divide both sides by 5,
    INT.[(e^x)cos(2x)]dx = (1/5)(e^x)cos(2x) +(2/5)(e^x)sin(2x) +C -----------answer.
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