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Math Help - integral by series

  1. #1
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    integral by series

    Thanks for any help.

    Assume that sin(x) equals its Maclaurin series for all x.
    Use the Maclaurin series for sin(3x^2) to evaluate the integral

    ∫sin(3x^2) from 0.6 to 0

    Your answer will be an infinite series. Use the first two terms to estimate its value.
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by waite3 View Post
    Thanks for any help.

    Assume that sin(x) equals its Maclaurin series for all x.
    Use the Maclaurin series for sin(3x^2) to evaluate the integral

    ∫sin(3x^2) from 0.6 to 0

    Your answer will be an infinite series. Use the first two terms to estimate its value.
    The Maclaurin series for sin(x) is
    sin(x) = \sum_{n = 0}^{\infty} \frac{(-1)^n}{(2n + 1)!}x^{2n + 1}

    So
    sin(3x^2) = \sum_{n = 0}^{\infty} \frac{(-1)^n}{(2n + 1)!}\left ( 3x^2 \right ) ^{2n + 1}

    Thus
    \int sin(3x^2)~dx = \int \sum_{n = 0}^{\infty} \frac{(-1)^n}{(2n + 1)!}\left ( 3x^2 \right ) ^{2n + 1}~dx

    = \sum_{n = 0}^{\infty} \frac{(-1)^n3^{2n+ 1}}{(2n + 1)!(4n + 3)}x^{4n + 3}

    I'll let you finish up from there.

    -Dan
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  3. #3
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    I have a couple questions; where do I plug in the 0.6 and how do I take into account the factorial.
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by waite3 View Post
    I have a couple questions; where do I plug in the 0.6 and how do I take into account the factorial.
    use the fundamental theoroem of calculus \int_a^{b}f(x)dx=F(b)-F(a),F'(x)=f(x)...so apply that with b=.6 and a=0(drops out no point in putting it in)
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  5. #5
    MHF Contributor Mathstud28's Avatar
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    Oh yeah and to account for the factorial I suppose you will have to use the gamma function haha \Gamma(x)=\int_0^{1}\bigg[ln\bigg(\frac{1}{t}\bigg)\bigg]^{x}dt
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  6. #6
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    i understand how to integrate the 0.6 now into the equation, but is there an easier way the solve this problem with the factorial in there
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  7. #7
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    problem is driving me nuts

    can anyone give me an exact answer cause the answers i keep getting are not right when i plug them into the program I'm using.

    thanks for any help
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  8. #8
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    i keep getting 1.07 something any help would be fantastic
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  9. #9
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by waite3 View Post
    i keep getting 1.07 something any help would be fantastic
    .199
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  10. #10
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    that seems not to be working either
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  11. #11
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by waite3 View Post
    that seems not to be working either
    \int_0^{.6}sin(3x^2)dx=.199=\sum_{n=0}^{\infty}\fr  ac{(-1)^{n}3^{2n+1}x^{4n+3}}{(4n+3)(2n)!}\bigg|_0^{.6}
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  12. #12
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    are there any additional decimals because sometime the program needs quite a few. thanks for the help.
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  13. #13
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by waite3 View Post
    are there any additional decimals because sometime the program needs quite a few. thanks for the help.
    0.19865856135
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