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Math Help - Integrate ∫e^7*sin(4x)

  1. #1
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    Integrate ∫e^7*sin(4x)

    Hi, I thought I had this question until I got to the very last part, I believe this question uses the repeated integration by parts method, is that right?

    Evaluate the indefinite integral:

    ∫e^7x*sin(4x)

    My work:
    u = e^7x
    u' = (1/7)e^7x
    v = (-1/4)cos(4x)
    v' = sin(4x)

    ∫e^7x*sin(4x) = (-1/4)e^7x*cos(4x) - ∫(-1/4)cos(4x)*(1/7)e^7x
    .....................= (-1/4)e^7x*cos(4x) + (1/28)
    ∫cos(4x)*e^7x <-- integrate this indefinite integral using integration by parts?

    So...
    (1/28) ∫cos(4x)*e^7x
    u = cos(4x)
    u' = (-1/4)sin(4x)
    v = (1/7)e^7x
    v' = e^7x

    And then I get...
    (1/28) ∫cos(4x)*e^7x = (1/28)(cos(4x)*(1/7)e^7x + ∫(1/7)e^7x*(-1/4)sin(4x))
    ..............................= (1/196)(cos(4x) + (1/28^2)∫e^7x*sin(4x) <-- so this second term is now the same as the original equation, except for the constant of course, and I should just simplify, right? But I don't know the fractions make it look really complicated and I don't think I'm doing it right...

    Does someone know where I went wrong and how to solve it properly?
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  2. #2
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    Re: Integrate ∫e^7*sin(4x)

    UBCBOY

    I FOUND e^(7x)[(7sin(4x)-4cos(4x))/65]+C
    it is more simple ..check it
    MINOAS
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  3. #3
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    Re: Integrate ∫e^7*sin(4x)

    Uhm... seriously, use LaTeX. Sorry, I didn't read all you wrote. Anyway, u'=7e^{7x}.

    I=\int e^{7x}\sin{4x}\, dx=\int \left (\frac{1}{7}e^{7x}  \right )' \sin{4x}\, dx=

    =\frac{1}{7}e^{7x}\sin{4x}-\int \frac{1}{7}e^{7x} (\sin{4x})'\, dx=\frac{1}{7}e^{7x}\sin{4x}-\int \frac{1}{7}e^{7x}\cdot  4\cos{4x}\, dx=

    =\frac{1}{7}e^{7x}\sin{4x}-\frac{4}{7}\int e^{7x} \cos{4x}\, dx=\frac{1}{7}e^{7x}\sin{4x}-\frac{4}{7}\int \left (\frac{1}{7}e^{7x}  \right )'  \cos{4x}\, dx  =

    =\frac{1}{7}e^{7x}\sin{4x}-\frac{4}{7}\left ( \frac{1}{7}e^{7x}\cos{4x}- \int \frac{1}{7}e^{7x} (\cos{4x})'\, dx  \right ) =

    =\frac{1}{7}e^{7x}\sin{4x}-\frac{4}{49}e^{7x}\cos{4x}+\frac{4}{49}\int e^{7x}\cdot 4(-\sin{4x})\,dx=

    =\frac{1}{7}e^{7x}\sin{4x}-\frac{4}{49}e^{7x}\cos{4x}-\frac{16}{49}\int e^{7x}\cdot \sin{4x}\,dx=

    =\frac{1}{7}e^{7x}\sin{4x}-\frac{4}{49}e^{7x}\cos{4x}-\frac{16}{49}I\Rightarrow

    \left ( 1+\frac{16}{49} \right )I=\frac{1}{7}e^{7x}\sin{4x}-\frac{4}{49}e^{7x}\cos{4x}\Rightarrow

    65I=7e^{7x}\sin{4x}-4e^{7x}\cos{4x}\Rightarrow I=e^{7x}\left ( \frac{7}{65}\sin {4x}-\frac{4}{65} \cos{4x} \right )+C
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  4. #4
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    Re: Integrate ∫e^7*sin(4x)

    Thank you guys. Is LaTex a program for solving calculus?
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  5. #5
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    Re: Integrate ∫e^7*sin(4x)

    Quote Originally Posted by UBCBOY View Post
    Thank you guys. Is LaTex a program for solving calculus?
    Hey UBCBOY!

    LaTeX is a typesetting format.
    If you click Reply With Quote on veileen's post, you'll see how it works.
    Basically you type your formulas like you'd usually would, then add [ TEX ] ... [ /TEX ] markers around it, and presto!
    A nicely formatted formula.
    If you click Go Advanced, you'll see a quick button with a \sum on it that facilitates this.
    Thanks from MarkFL
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