\(\int \frac{\sin 2 x\left(1-\frac{3}{2} \cos x\right)}{\mathrm{e}^{\sin ^2 x+\cos ^3 x}} \mathrm{~d} x=\)

\(\text { Put } \sin ^2 x+\cos ^3 x=\mathrm{t} \)
\( \Rightarrow\left(2 \sin x \cos x-3 \cos ^2 x \sin x\right) \mathrm{d} x=\mathrm{dt} \)
\( \Rightarrow\left(\sin 2 x-\frac{3}{2} \sin 2 x \cos x\right) \mathrm{d} x=\mathrm{dt} \)
\( \Rightarrow \sin 2 x\left(1-\frac{3}{2} \cos x\right) \mathrm{d} x=\mathrm{dt} \)
\( \therefore \int \frac{\sin 2 x\left(1-\frac{3}{2} \cos x\right)}{\mathrm{d}} \mathrm{d} x \)
\( =\int \frac{1}{\mathrm{e}^{\mathrm{t}}} \mathrm{dt}^2 x+\cos ^3 x \)
\( =\int \mathrm{e}^{-\mathrm{t}} \mathrm{dt} \)
\( =-\mathrm{e}^{-\mathrm{t}}+\mathrm{c} \)
\( =-\mathrm{e}^{-\left(\sin ^2 x+\cos ^3 x\right)}+\mathrm{c}\)