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Math Help - Integral Divergence or Convergence Using Comparison Theorm

  1. #1
    Member zangestu888's Avatar
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    Integral Divergence or Convergence Using Comparison Theorm

    The question is;
    Use the Comparison theorm to determine wheater the integral is convergent or divergent.

    I) integral (cosx)^2*dx/1+x^2 a=1 b=infinte

    II) integral dx/x+ e^2x a=1 b=infinte

    Answers are I=Convergent, II=convergent

    Help Much Appreciated!
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Here are some hints
    Quote Originally Posted by zangestu888 View Post
    Use the Comparison theorm to determine wheater the integral is convergent or divergent.

    I) integral (cosx)^2*dx/1+x^2 a=1 b=infinte
    since 0 \le \cos^2 x \le 1, we can compare to \int_1^\infty \frac 1{x^2}~dx

    II) integral dx/x+ e^2x a=1 b=infinte
    since e^{2x} \gg x we can compare to \int_1^\infty \frac 1{e^{2x}}~dx
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  3. #3
    Member zangestu888's Avatar
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    okay first one am okay with but will 1/e^2x converge as well?
    o true doing the integral it will converge to 1/2e^-2 okay thanks!
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by zangestu888 View Post
    okay first one am okay with but will 1/e^2x converge as well?
    o true doing the integral it will converge to 1/2e^-2 okay thanks!
    that's right, you can work it out directly or note that we also have \frac 1{e^{2x}} \le \frac 1{x^2} for x \ge 1, so another comparison would work
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