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Math Help - Partial Integration

  1. #1
    Junior Member
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    Johannesburg, South Africa
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    Partial Integration

    I have \alpha:[0,T]\rightarrow \mathbb{R} is continuous and \beta>0
    How do I apply partial integration on \alpha(t)+\beta\int_0^t e^{\beta(t-s)}\alpha(s)ds such that it equals
    e^{\beta t}\alpha(0)+\int_0^te^{\beta(t-s)}\alpha'(s)ds
    Thanks
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  2. #2
    Super Member Random Variable's Avatar
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    let  u = \alpha (s) and  dv = e^{\beta (t-s)} ds

     \displaystyle = \alpha (t) - \alpha(s) e^{\beta(t-s)} \Big|^{t}_{0} + \int^{t}_{0} e^{\beta(t-s)} \alpha^{'}(s) \ ds

     \displaystyle = \alpha (t) - \alpha(t) + \alpha(0) e^{\beta t}  + \int^{t}_{0} e^{\beta(t-s)} \alpha^{'}(s) \ ds
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