In the first integral, substitute [tex]x \to e^{\sqrt{e^x}}[/tex]:
[tex]\displaystyle I = \int_0^1 e^{\sqrt{e^x}} \, dx = 2 \int_e^{e^{\sqrt e}} \frac{dx}{\ln(x)}[/tex]
In the second integral, integrate by parts:
[tex]\displaystyle J = \int_e^{e^{\sqrt e}} \ln(\ln(x)) \, dx = \dfrac12 e^{\sqrt e} - \int_e^{e^{\sqrt e}} \frac{dx}{\ln(x)}[/tex]
It follows that
[tex]\dfrac{10^5}{e^{\sqrt e}}(I+2J) = \dfrac{10^5}{e^{\sqrt e}} \times e^{\sqrt e} = \boxed{10^5}[/tex]
