#### Answer

$y(t)=-\cos (e^t-2) +1$

#### Work Step by Step

Given: $\dfrac{dy}{dt}=e^t \sin (e^t-2)$
Here, $y=\int e^t \sin (e^t-2) dt$
Plug $a=e^t-2 \implies da=e^t dt$
Now, we have $y=\int \sin a da$
or, $y=-\cos a da+c$
Thus, $y=-\cos (e^t-2 )+c$
Apply initial conditions $y[\ln (2)]=2$
Thus, we have $y(t)=-\cos (e^t-2) +1$