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Math Help - integral/sin proof

  1. #1
    Junior Member
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    integral/sin proof

    prove that the integral from 0 to x of the function [sin(t)]/(t+1) with respect to t is greater than 0 for all x>0

    i dont know how to do the math script but it looks something like this
    subscript (0) superscript (x) integral([sin(t)]/(t+1))dt >0 for all x>0
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  2. #2
    Math Engineering Student
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    More generally: consider f a continuously differentiable function, positive and decreasing for (0,\infty) then \int_0^x f(t)\sin t\,dt>0.

    Proof: integrate by parts and get \int_{0}^{x}{f(t)\sin t\,dt}=\bigg. (1-\cos t)f(t) \bigg|_{0}^{x}-\int_{0}^{x}{(1-\cos t)f'(t)\,dt}.

    Now 1-\cos t>0 and f was given as positive, hence (1-\cos x)f(x)>0, on the other hand f was given as decreasing, so f'(t)<0, thus the whole RHS is positive, as claimed. \blacksquare
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