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Math Help - Limits involving integrals

  1. #1
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    Limits involving integrals

    how can i solve by FTC?
    <br />
\underset{x\to 0}{\mathop{\lim }}\,\frac {1}{x^{2}}\int_{0}^{x^{2}}{\frac {t + \sin ^{2}\sqrt {t}}{t + 1}}dt

    same condition for this one (use FTC)
    find F'(x)
    <br />
F\left( x \right) = \int_{0}^{x}{\ln \left( (tx)^{x} \right)}dt

    and this one too :roll:
    find
    g''\left( x \right) + g\left( x \right)
    where
    <br />
g\left( x \right) = \int_{0}^{x}{f(t)\sin (x)dt}<br />
    Last edited by mr fantastic; March 18th 2010 at 01:55 AM. Reason: Changed post title
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by vivancos View Post
    how can i solve by FTC?
    <br />
\underset{x\to 0}{\mathop{\lim }}\,\frac {1}{x^{2}}\int_{0}^{x^{2}}{\frac {t + \sin ^{2}\sqrt {t}}{t + 1}}dt

    same condition for this one (use FTC)
    find F'(x)
    <br />
F\left( x \right) = \int_{0}^{x}{\ln \left( (tx)^{x} \right)}dt

    and this one too :roll:
    find
    g''\left( x \right) + g\left( x \right)
    where
    <br />
g\left( x \right) = \int_{0}^{x}{f(t)\sin (x)dt}<br />
    Hint: Apply L'Hopital's rule.
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  3. #3
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    Quote Originally Posted by chiph588@ View Post
    Hint: Apply L'Hopital's rule.
    can u solve it please
    Last edited by vivancos; March 17th 2010 at 07:56 AM.
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  4. #4
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    \lim_{x\to 0} \, \frac{\int_{0}^{x^{2}}{\frac {t + \sin ^{2}\sqrt {t}}{t + 1}}dt}{x^2}

    note that direct substitution of x = 0 leads to the indeterminate form \frac{0}{0}.

    use L'Hopital's rule ...

    \lim_{x\to 0} \frac{\frac{d}{dx}\left[\int_{0}^{x^{2}}{\frac {t + \sin ^{2}\sqrt {t}}{t  + 1}}dt\right]}{\frac{d}{dx}(x^2)} = \lim_{x\to 0} \, \frac{\frac{x^2 + \sin^2{|x|}}{x^2 + 1} \cdot 2x}{2x} = \lim_{x\to 0} \, \frac{x^2 + \sin^2{|x|}}{x^2 + 1} = 0
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