The task is to integrate using substitution:
$$\int\sin(2x)\cdot \frac{(e^{\cos^{2} x})}{({e^{(\sin^2 x)}})}$$
I tried choosing $u= (e^{(\sin x)^2}) $and then $dx=du/\sin(2x)\cdot(e^{(\sin x)^2})$
This simplified it to $(e^{(\cos x)^2})/(e^{(\sin x)^2}) \cdot 1/u$ and since $u=(e^{(\sin x)^2})$ I replaced $(e^{(\sin x)^2})$ with $u$:
$(e^{(\cos x)^2}) \cdot 1/u^2$
But I'm not sure what to do next...
HINT:
So, we have $$\int e^{\cos^2x-\sin^2x}\sin2x\ dx$$
$$\dfrac{d(\cos^2x-\sin^2x)}{dx}=?$$
Also $\cos^2x-\sin^2x=\cos2x$