If $I_1=\underset{0}{\overset{1}{\int }}{e}^{-x}{\cos }^{2}xdx,I_2=\underset{0}{\overset{1}{\int }}{e}^{-x^2}{\cos }^{2}xdx$ and $I_3=\underset{0}{\overset{1}{\int }}{e}^{-x^3}dx,$ then 

$$\begin{gathered} {\mathrm{I}}_1={\int }_0^1 {\mathrm{e}}^{-\mathrm{x}} {\cos }^2\mathrm{x}.\mathrm{dx}, {\mathrm{I}}_2={\int }_0^1 {\mathrm{e}}^{-{\mathrm{x}}^2} {\cos }^2\mathrm{x} \mathrm{dx} \\ {\mathrm{I}}_3={\int }_0^1 {\mathrm{e}}^{-{\mathrm{x}}^3}\mathrm{dx} \\ 0<\mathrm{x}<1 \Rightarrow \mathrm{x}>{\mathrm{x}}^2>{\mathrm{x}}^3 \\ \Rightarrow -\mathrm{x}<-{\mathrm{x}}^2<-{\mathrm{x}}^3 \\ \Rightarrow {\mathrm{e}}^{-\mathrm{x}}<{\mathrm{e}}^{-{\mathrm{x}}^2}<{\mathrm{e}}^{-{\mathrm{x}}^3} \\ \mathrm{Since} {\mathrm{e}}^{-\mathrm{x}}<{\mathrm{e}}^{-{\mathrm{x}}^2} \\ \Rightarrow {\mathrm{e}}^{-\mathrm{x}}{\cos }^2\mathrm{x}<{\mathrm{e}}^{-{\mathrm{x}}^2}{\cos }^2\mathrm{x} \\ \Rightarrow {\int }_0^1 {\mathrm{e}}^{-\mathrm{x}}{\cos }^2\mathrm{x} \mathrm{dx}<{\int }_0^1 {\mathrm{e}}^{-{\mathrm{x}}^2} {\cos }^2\mathrm{x} \mathrm{dx} \\ {\mathrm{I}}_1<{\mathrm{I}}_2 \end{gathered}$$

$$\begin{gathered} \mathrm{and} {\cos }^2\mathrm{x}<1 \\ \mathrm{and} {\mathrm{e}}^{-{\mathrm{x}}^2}<{\mathrm{e}}^{-{\mathrm{x}}^3} \\ \Rightarrow {\mathrm{e}}^{-{\mathrm{x}}^2{\cos }^2\mathrm{x}<{\mathrm{e}}^{-{\mathrm{x}}^3}} \\ {\int }_0^1 {\mathrm{e}}^{-{\mathrm{x}}^2}{\cos }^2\mathrm{x} \mathrm{dx}<{\int }_0^1 {\mathrm{e}}^{-{\mathrm{x}}^3}\mathrm{dx} \\ {\mathrm{I}}_2<{\mathrm{I}}_3 \\ {\mathrm{I}}_1<{\mathrm{I}}_2<{\mathrm{I}}_3 \end{gathered}$$