Question

`\rm \frac{ {10}^5 }{ {e}^( √(e) ) } \left( \int^(1)_0 {e}^{ \sqrt[]{ {e}^(x) } } \: dx + 2 \int_(e)^{ {e}^( √(e) ) } ln( ln(x) ) \: dx\right) \\`​

Answer 1

In the first integral, substitute
`x \to e^(√(e^x))`:

`\displaystyle I = \int_0^1 e^(√(e^x)) \, dx = 2 \int_e^{e^(\sqrt e)} (dx)/(\ln(x))`

In the second integral, integrate by parts:

`\displaystyle J = \int_e^{e^(\sqrt e)} \ln(\ln(x)) \, dx = \frac12 e^(\sqrt e) - \int_e^{e^(\sqrt e)} (dx)/(\ln(x))`

It follows that

`(10^5)/(e^(\sqrt e))(I+2J) = (10^5)/(e^(\sqrt e)) * e^(\sqrt e) = \boxed{10^5}`