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Math Help - integral question

  1. #1
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    integral question

    h(x)= integral from 0 to e^x of ln(t) dt

    solution

    let u = e^x then du/dx =e^x

    also dh/dx=dh/du du/dx

    h'(x)=d/dx integral from 1 to e^x of ln(t) dt = d/du integral from 1 to u ln(t) dt times du/dx = ln u du/dx = (lne^x)*e^x = xe^x

    for "d/du integral from 1 to u ln(t) dt times du/dx" above, why is the integral multiplied by the derivative of u WRT x
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  2. #2
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    Re: integral question

    Quote Originally Posted by kingsolomonsgrave View Post
    [FONT=Times New Roman][SIZE=2][COLOR=#0e0e0e][FONT=Times New Roman][SIZE=2][COLOR=#0e0e0e]h(x)= integral from 0 to e^x of ln(t) dt

    Do you really mean \int_0^{e^x } {\ln (t)dt} or was that a typo and you want \int_1^{e^x } {\ln (t)dt}~?
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  3. #3
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    Re: integral question

    the second one, sorry
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    Re: integral question

    \frac{d}{dx} \int_a^u f(t) \, dt = f(u) \cdot \frac{du}{dx}

    \frac{d}{dx} \int_1^{e^x} \ln(t) \, dt = \ln(e^x) \cdot e^x = x \cdot e^x
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  5. #5
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    Re: integral question

    Quote Originally Posted by kingsolomonsgrave View Post
    the second one, sorry

    If h(x)=\int_1^{e^x } {\ln (t)dt} then h'(x)=xe^x~.

    Because \ln(e^x)=x and the derivative of e^x=e^x~.
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