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Math Help - Indefinite and Definite integrals as Power series

  1. #1
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    Indefinite and Definite integrals as Power series

    F(x) = integral of [x / (1-x^6)]

    Integral from 0 to .3 of [Ln (1+x^3)] dx

    I'm so lousy at this stuff. I don't know how to even start these.

    And thanks to everyone that helped me before. I really appreciate it. I'm not just looking for answers. I'm really trying to do these... I just CAN'T get it right.
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  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Quote Originally Posted by thegame189 View Post
    F(x) = integral of [x / (1-x^6)]

    Integral from 0 to .3 of [Ln (1+x^3)] dx

    I'm so lousy at this stuff. I don't know how to even start these.

    And thanks to everyone that helped me before. I really appreciate it. I'm not just looking for answers. I'm really trying to do these... I just CAN'T get it right.
    here is the first one

    let g(x)=\frac{1}{1-x}=\sum_{n=0}^{\infty}x^n
    then
    g(x^6)=\frac{1}{1-x^6}=\sum_{n=0}^{\infty}x^{6n}

    finally

    F(x)=\frac{x}{1-x^6}=xg(x^6)=x\sum_{n=0}^{\infty}x^{6n}=\sum_{n=0}  ^{\infty}x^{6n+1}

    integraing both ends gives

     \int F(x)dx = \int \sum_{n=0}^{\infty} x^{6n+1} dx = C+ \sum_{n=0}^{\infty} \frac{x^{6n+2}}{6n+2}
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  3. #3
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Quote Originally Posted by thegame189 View Post
    F(x) = integral of [x / (1-x^6)]

    Integral from 0 to .3 of [Ln (1+x^3)] dx

    I'm so lousy at this stuff. I don't know how to even start these.

    And thanks to everyone that helped me before. I really appreciate it. I'm not just looking for answers. I'm really trying to do these... I just CAN'T get it right.
    Hint let g(x)=\ln(1+x)

    then g'(x)=\frac{1}{1+x}

    use the same proceedure as above.

    Good luck.
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