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Math Help - Integration questions

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    Integration questions

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    Quote Originally Posted by champrock View Post
    For the first one, let t = a-s so the integral becomes

    S = -e^a \int_{a-1}^a \frac{e^{-s}}{s-a-1}\,ds

    or if your replace s with t in mine and then solve for the integral (note - you're missing a dt)

    For the second, let t = e^{x^2} (the hint is in how the limits of integration change) so the given integral becomes

    \alpha = \frac{1}{2} \int_e^{e^4} \frac{dt}{\sqrt{\ln t}}

    now we'll integrate by parts

    \frac{1}{2}\int_e^{e^4} \frac{t\,dt}{t\,\sqrt{\ln t}} with  u = t,\;dv = \frac{1}{t\,\sqrt{\ln t}}

    so

    \frac{1}{2} \int_e^{e^4} \frac{t\,dt}{t\,\sqrt{\ln t}} = \left. t \sqrt{\ln t}\right|_e^{e^4} - \int_e^{e^4} \sqrt{\ln t}\, dt = 2e^4 - e - \int_e^{e^4} \sqrt{\ln t}\, dt = \alpha and solve for your integral.
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    Hi

    Got the first question.

    can u please elaborate how t=e^x^2 becomes this:
    <br />
\alpha = \frac{1}{2} \int_e^{e^4} \frac{dt}{\sqrt{\ln t}}<br />
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    Quote Originally Posted by champrock View Post
    Hi

    Got the first question.

    can u please elaborate how t=e^x^2 becomes this:
    <br />
\alpha = \frac{1}{2} \int_e^{e^4} \frac{dt}{\sqrt{\ln t}}<br />
    t = e^{x^2} \Rightarrow \frac{dt}{dx} = 2x e^{x^2} \Rightarrow dx = \frac{dt}{2x e^{x^2}}.


    Therefore \int_1^2 e^{x^2} \, dx = \int_e^{e^4} \frac{ e^{x^2} dt}{2x e^{x^2}} = \frac{1}{2} \int_e^{e^4} \frac{dt}{x}.


    But t = e^{x^2} \Rightarrow \ln t = x^2 \Rightarrow \sqrt{\ln t} = x. Substitute this into the above.
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