1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Integral convergence

  1. Feb 25, 2008 #1
    1. The problem statement, all variables and given/known data

    [tex]\int[/tex][tex]^{\infty}_{}[/tex] (cos x)/(x +exp(x)) dx
    [tex]_{ 0}[/tex]

    [tex]\int[/tex][tex]^{\infty}_{}[/tex] (x + [tex]\sqrt{x})^{-1}[/tex]dx
    [tex]_{ 1}[/tex]

    [tex]\int[/tex][tex]^{\infty}_{}[/tex] (1 + x[tex]^{3}[/tex])[tex]^{-1/2}[/tex] dx
    [tex]_{ 1}[/tex]

    3. The attempt at a solution

    i) x +exp(x) [tex]\geq[/tex] 1

    -1[tex]\leq[/tex]cos x [tex]\leq[/tex]1

    -1[tex]\leq[/tex](cos x)/(x +exp(x))[tex]\leq[/tex]1/(x +exp(x))

    then do i have to compare something to something knowing for certain that something convereges?

    For ii) + iii) please can someone nudge me in the right direction.
    Last edited: Feb 25, 2008
  2. jcsd
  3. Feb 25, 2008 #2
    for the first one use the fact that cosx<1 for x from zero to infinity, also use the fact that 1/(x+exp(x))<exp(-x), then try to show that exp(-x) converges, so there is a theorem i guess, i am not sure how exactly it goes but i think it say that if

    f(x)<g(x), then also

    integ (from a to x)f(x)dx<integ(from a to x) g(x), then if the right hand sided integral converges say to a nr M, then it means that integ (from a to x)f(x)dx<M, so it means that this function is upper bounded so it also must have a precise upper bound, hence the limit also must exist as x-->infinity, which actually tells us that the integral
    [tex]\int_0^{\infty} \frac{cos x}{x+e^{x}}dx[/tex] converges
    Last edited: Feb 25, 2008
  4. Feb 25, 2008 #3
    For the other two i believe you can find the antiderivatives of those functions, and see whether they converge or not!

    [tex]\int_0^{\infty}\frac{dx}{x+\sqrt x}=\lim_{b\rightarrow\infty} \int_0^{b}\frac{dx}{x+\sqrt x}[/tex], now let [tex]x=t^{2} => dx=2tdt,
    t=\sqrt x[/tex] [tex]x=0 =>t=0, when, x=b => t=\sqrt b[/tex]
    [tex]\lim_{b\rightarrow\infty} \int_0^{\sqrt b}\frac{2tdt}{t^{2}+t}=2\lim_{b\rightarrow\infty}\int_0^{\sqrt b}\frac{dt}{1+t}[/tex],
    Last edited: Feb 26, 2008
  5. Feb 25, 2008 #4
    I think also for the iii) you will be able to find an antiderivative in terms of an el. function!
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Integral convergence
  1. Convergent integral (Replies: 1)