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Math Help - Integrals

  1. #1
    Junior Member
    Joined
    Dec 2008
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    30

    Integrals

    does the integral <br />
\int_0^1 {\frac{{e^x }}{{\sqrt {1 - \cos (x)} }}} <br /> <br />
converge or diverge???

    and if you can show how do I see it!
    thanks ahead
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  2. #2
    Math Engineering Student
    Krizalid's Avatar
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    Apply limit comparison test with \int_0^1\frac{dx}{\sqrt x}, which converges, hence \underset{x\to 0}{\mathop{\lim }}\,\frac{{{e}^{x}}}{\sqrt{1-\cos x}}\div \frac{1}{\sqrt{x}}=\underset{x\to 0}{\mathop{\lim }}\,\frac{\sqrt{x}{{e}^{x}}}{\sqrt{1-\cos x}}=\underset{x\to 0}{\mathop{\lim }}\,\sqrt{\frac{x}{1-\cos x}}\cdot {{e}^{x}}=0, since \frac{x}{1-\cos x}\to0 as x\to0. Finally, the integral converges.
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