For a differentiable continuous function $g(u)$ over a finite region $R$, I know that $$\int_R e^u g\; u\; du = 1$$ Is there any way to determine from this the value of $$\int_R 10^u g\; u\; du$$
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1What is $g$ alone. – hamam_Abdallah Nov 01 '16 at 19:37
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We don't have any expression for $g(u)$; only that it's the same in both integrals, as is the range of integration. – Oliver Nov 02 '16 at 08:49
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It is possible only if $g(u)$=constant or linear. – hamam_Abdallah Nov 02 '16 at 09:03
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The second integral is
$$J=\int 10^u g(u)du =$$
$$=\int e^{uln(10)}g(u)du=.$$
put $v=u ln(10)$
then
$$J=\frac{1}{ln(10)}\int e^v g(\frac{v}{ln(10)})dv.$$
So, in general, we cannot get tha value of $J$
from $I=\int e^u g(u)du=1$.
if $g$ is linear then $J=\frac{1}{(ln(10))^2}$.
hamam_Abdallah
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No, it depends on $g$ look at the two cases $g(x)=x$ and $g(x)=x^2$.
Rene Schipperus
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How do I look at these? What is known is the value of the integral. The function g is identical for the two integrals shown; the only difference is $e^u$ vs $10^u$. – Oliver Nov 02 '16 at 08:49