Communicable $e$

$\begin{aligned} I & = \int_0^{\frac{\pi}{2}} \sin \left(\sqrt{12e^{\sqrt{e^{\ln (100) - \ln (4)}}}} \right) \text{d} \theta & \small \color{#3D99F6} \ln (100) - \ln(4) = \ln \left(\frac{100}{4}\right) \\ & = \int_0^{\frac{\pi}{2}} \sin \left(\sqrt{12e^{\sqrt{e^{\ln (25) }}}} \right) \text{d} \theta & \small \color{#3D99F6} e^{\ln (25)} = 25 \\ & = \int_0^{\frac{\pi}{2}} \sin \left(\sqrt{12e^{\sqrt{25}}} \right) \text{d} \theta & \small \color{#3D99F6} \sqrt{25} = 5 \\ & = \int_0^{\frac{\pi}{2}} \sin \left(\sqrt{12e^{5}} \right) \text{d} \theta & \small \color{#3D99F6} \sqrt{12e^5} = \sqrt{12}\sqrt{e^5} = 2\sqrt{3}e^{\frac 52} \\ & = \int_0^{\frac{\pi}{2}} \sin \left(2\sqrt{3}e^{\frac 52} \right) \text{d} \theta \\ & = \sin \left(2\sqrt{3}e^{\frac 52} \right) \theta \left. \right|_0^{\frac{\pi}{2}} \\ & = \frac{\pi}{2} \sin\left(2\sqrt{3}e^{\frac 52}\right) - 0\sin\left(2\sqrt{3}e^{\frac 52}\right) \\ & = \boxed{\dfrac{\pi}{2} \sin\left(2\sqrt{3}e^{\frac 52}\right)} \end{aligned}$