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Math Help - Help with Limit and Integration

  1. #1
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    Help with Limit and Integration

    1. Find the limit: lim x->0

    (cos 3x)^1/x (Hint: use logarithmic method)


    2. let f(x)=(5x^2 + 1)/((x-1)(x+1)^2) Find a,b,c such that f(x)=(a/(x-1))+(b/(x+1))+(c/(x+1)^2), and use the decomposition to evaluate the integral /f(x)dx

    3. Evaluate improper integral: from 0-infinity xe^-x dx(hint: use integration by parts)

    4. Evaluate improper integral: from0-2 (dx/(sqrt(4-x^2))

    5. The region R enclosed by the curves y-x/4 and y^2=x is rotated about the y-axis. Find the volume of the resulting solid.


    Thank you very much for your help
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  2. #2
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    Quote Originally Posted by drmguy View Post
    1. Find the limit: lim x->0

    (cos 3x)^1/x (Hint: use logarithmic method)


    2. let f(x)=(5x^2 + 1)/((x-1)(x+1)^2) Find a,b,c such that f(x)=(a/(x-1))+(b/(x+1))+(c/(x+1)^2), and use the decomposition to evaluate the integral /f(x)dx

    3. Evaluate improper integral: from 0-infinity xe^-x dx(hint: use integration by parts)

    4. Evaluate improper integral: from0-2 (dx/(sqrt(4-x^2))

    5. The region R enclosed by the curves y-x/4 and y^2=x is rotated about the y-axis. Find the volume of the resulting solid.


    Thank you very much for your help
    3. I don't like integration by parts, I prefer to use the "go in reverse twice" method.

    \frac{d}{dx}(xe^{-x}) = e^{-x} - xe^{-x}

    So \int{e^{-x} - xe^{-x}\,dx} = xe^{-x} + C

    \int{e^{-x}\,dx} - \int{xe^{-x}\,dx} = xe^{-x} + C

    -e^{-x} - \int{xe^{-x}\,dx} = xe^{-x} + C

    \int{xe^{-x}\,dx} = -xe^{-x} - e^{-x} + C


    Now to evaluate this integral from 0 to \infty...

    \int_0^\infty{xe^{x}\,dx} = \lim_{\varepsilon \to \infty}(-\varepsilon e^{-\varepsilon} - e^{-\varepsilon} + C) - (-0e^{-0} - e^{-0} + C)

    Can you go from there?
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  3. #3
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    Quote Originally Posted by drmguy View Post
    1. Find the limit: lim x->0

    (cos 3x)^1/x (Hint: use logarithmic method)


    2. let f(x)=(5x^2 + 1)/((x-1)(x+1)^2) Find a,b,c such that f(x)=(a/(x-1))+(b/(x+1))+(c/(x+1)^2), and use the decomposition to evaluate the integral /f(x)dx

    3. Evaluate improper integral: from 0-infinity xe^-x dx(hint: use integration by parts)

    4. Evaluate improper integral: from0-2 (dx/(sqrt(4-x^2))

    5. The region R enclosed by the curves y-x/4 and y^2=x is rotated about the y-axis. Find the volume of the resulting solid.


    Thank you very much for your help
    4. \int_0^2{\frac{1}{\sqrt{4- x^2}}\,dx} = \left[\arcsin{\left(\frac{x}{2}\right)}\right]_0^2

     = \lim_{\varepsilon \to 2}\left[\arcsin{\left(\frac{\varepsilon}{2}\right)}\right] - \arcsin{\left(\frac{0}{2}\right)}

    Can you go from here?
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  4. #4
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    I am just worried that if i dont use integration by parts, as he requested, then i wont get full credit.
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