Answer
$1$
Work Step by Step
Our aim is to solve the integral $ \int_{\ln(\pi/2)}^{\ln(\pi)} e^x \sin e^x \ dx$
Let us consider that $a =e^x$ and $\ da=e^x \ dx$
Now, $\int_{\ln(\pi/2)}^{\ln(\pi)} e^x \sin e^x \ dx = \int \sin a \ da$
or, $=-\cos a +C$
or, $=-\cos(e^x)+C$
Now, $\int_{\ln(\pi/2)}^{\ln(\pi)} e^x \sin e^x \ dx =-[\cos (e^x)]_{\ln(\pi/2)}^{\ln(\pi)}\\=-[\cos (e^{\ln \pi})-\cos (e^{\ln (\pi/2)}]\\=-[-1-0]\\=1$
