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Math Help - Another Improper Integral

  1. #1
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    Another Improper Integral

    I'm trying to show that \int_{-2}^{14}\frac{dx}{\sqrt[4]{x+2}}=<br />
\frac{32}{3}

    Since the integrand \frac{1}{\sqrt[4]{x+2}} is continuous on (-2,14] I can define

    \int_{-2}^{14}\frac{dx}{\sqrt[4]{x+2}}=\lim_{t->-2^+}\int_{t}^{14}\frac{dx}{\sqrt[4]{x+2}}

    =\lim_{t->-2^+}\int_{t}^{14}(x+2)^{-\frac{1}{4}}dx

    =\lim_{t->-2^+}-\frac{4}{3\sqrt[4]{(x+2)^4}}\displaystyle{]_{x=t}^{x=14}}

    =-\frac{1}{4}+\lim_{t->-2^+}\frac{4}{3\sqrt[4]{(t+2)^3}}

    I'm stuck on the limit. If I didn't know any better, I would say that the integral diverges. I tried writing \lim_{t->-2^+}\frac{4\sqrt[4]{(t+2)^3}}{3(t+2)^3} so I could use L-hopitals rule, but it doesn't seem to go anywhere since the derivative of the numerator always give me a function of the form \frac{c}{f(t)} where f(-2)=0. So what's the trick here?
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by adkinsjr View Post
    I'm trying to show that \int_{-2}^{14}\frac{dx}{\sqrt[4]{x+2}}=<br />
\frac{32}{3}

    Since the integrand \frac{1}{\sqrt[4]{x+2}} is continuous on (-2,14] I can define

    \int_{-2}^{14}\frac{dx}{\sqrt[4]{x+2}}=\lim_{t->-2^+}\int_{t}^{14}\frac{dx}{\sqrt[4]{x+2}}

    =\lim_{t->-2^+}\int_{t}^{14}(x+2)^{-\frac{1}{4}}dx

    =\lim_{t->-2^+}-\frac{4}{3\sqrt[4]{(x+2)^4}}\displaystyle{]_{x=t}^{x=14}}

    =-\frac{1}{4}+\lim_{t->-2^+}\frac{4}{3\sqrt[4]{(t+2)^3}}

    I'm stuck on the limit. If I didn't know any better, I would say that the integral diverges. I tried writing \lim_{t->-2^+}\frac{4\sqrt[4]{(t+2)^3}}{3(t+2)^3} so I could use L-hopitals rule, but it doesn't seem to go anywhere since the derivative of the numerator always give me a function of the form \frac{c}{f(t)} where f(-2)=0. So what's the trick here?
    \lim_{t\to-2^+}\int_{t}^{14} (x+2)^{-1/4}\,dx=\lim_{t\to-2^+}\tfrac{4}{3}\left.\left[(x+2)^{3/4}\right]\right|_{t}^{14}.

    I'm sure you can take it from here...
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  3. #3
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    oh ok, no wonder I was having such a hard time with this one. I don't know why I decided that the exponent was negative. Thanks for your help.
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