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Math Help - Conditional convergence of improper integrals

  1. #1
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    Conditional convergence of improper integrals

    What is the integral of cos t/t where limits are 1 to x ?

    My book says that it is sinx/x - sin1 + integral (from 1 to x) sin t/ t2 dt for all x>=1.

    I can't figure out how .


    Please help.
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  2. #2
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    Re: Conditional convergence of improper integrals

    Quote Originally Posted by mrmaaza123 View Post
    What is the integral of cos t/t where limits are 1 to x ?

    My book says that it is sinx/x - sin1 + integral (from 1 to x) sin t/ t2 dt for all x>=1.

    I can't figure out how .

    Please help.

    Use integration by parts.

    u=\frac{1}{t} \implies du=-\frac{1}{t^2}dt

    and

    dv=\cos(t) \implies v=\sin(t)

    This gives

    \int_{1}^{x} \frac{\cos(t)}{t}dt=\frac{\sin(t)}{t}\bigg|_{t=1}^  {t=x}+\int_{1}^x\frac{\sin(t)}{t^2}dt
    Thanks from mrmaaza123
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  3. #3
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    Re: Conditional convergence of improper integrals

    Oh, yes. Thank you, i don't really know how i overlooked that !
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