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Math Help - Beginning calculus trig-related integral

  1. #1
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    Beginning calculus trig-related integral

    So it starts like this:

    Find the derivative of y = isinx + cosx

    And I don't know what I'm supposed to do with the complex number so I just sort of left it and got this:

    dy/dx = -sinx + icosx
    Which is equal to i * y


    Then it says to
    1) find dy/y
    2) use the condition x=0 to determine the value of c (constant of integration)
    3) write this antiderivative in its equivalent form

    I don't know what to do next! Please dumb it down as much as possible.
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  2. #2
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    Re: Beginning calculus trig-related integral

    Since you have \displaystyle \begin{align*} \frac{dy}{dx} = i\,y \end{align*}, what does that mean \displaystyle \begin{align*} \frac{dy}{y} \end{align*} is?
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  3. #3
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    Re: Beginning calculus trig-related integral

    idx.
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  4. #4
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    Re: Beginning calculus trig-related integral

    Yes, so now that you have \displaystyle \begin{align*} \frac{dy}{y} = i\,dx \end{align*} you should be able to integrate both sides.
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    Re: Beginning calculus trig-related integral

    ln(y) = ∫idx
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    Re: Beginning calculus trig-related integral

    First of all, \displaystyle \begin{align*} \int{\frac{dy}{y}} = \ln{|y|} \end{align*}, not \displaystyle \begin{align*} \ln{(y)} \end{align*}. And what would be \displaystyle \begin{align*} \int{i\,dx} \end{align*}? Remember that \displaystyle \begin{align*} i \end{align*} is a constant.
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  7. #7
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    Re: Beginning calculus trig-related integral

    ln|y| = ix + c
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    Re: Beginning calculus trig-related integral

    So boundary condition x=0
    y = isin(0) + cos(0)
    y = 1

    ln|1| = i(0) + c
    c = 0

    Is this right? What does it mean to write the antiderivative in its equivalent form?
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  9. #9
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    Re: Beginning calculus trig-related integral

    So if
    ln|y| = ix + 0
    then would the equivalent form be
    y = e^(ix)
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  10. #10
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    Re: Beginning calculus trig-related integral

    Quote Originally Posted by euphony View Post
    So if
    ln|y| = ix + 0
    then would the equivalent form be
    y = e^(ix)
    That is the correct equivalent form, however, your elimination of the constant is incorrect due to the absolute value around the y.

    \displaystyle \begin{align*} \ln{|y|} &= i\,x + C \\ |y| &= e^{i\,x + C} \\ |y| &= e^C e^{i\,x} \\ y &= A\,e^{i\,x} \end{align*}

    Now use your boundary condition to evaluate this constant A.
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  11. #11
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    Re: Beginning calculus trig-related integral

    y = Ae^(ix)
    (1) = A * e^[i(0)]

    A = 1?
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  12. #12
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    Re: Beginning calculus trig-related integral

    Yes much better
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