Mathematics - Definite Integration Question with Solution | TestHub

Given: $$\begin{array}{rl}& I_1={\int }_0^1{e}^{-x}{\cos }^{2}xdx\\ & I_2={\int }_0^1{e}^{-x^2}{\cos }^{2}xdx\text{ and }\\ & I_3={\int }_0^1{e}^{-x^3}dx\\ & \text{ For }x\in (0,1)\\ & \Rightarrow x>x^2\text{ or }-x<-x^2\\ & \text{ and }{x}^2>x^3\text{ or }-x^2<-x^3\\ & \therefore e^{-x^2}<e^{-x^3}\text{ and }{e}^{-x}<e^{-x^2}\\ & \Rightarrow e^{-x}<e^{-x^2}<e^{-x^3}\\ & \Rightarrow e^{-x^3}>e^{-x^2}>e^{-x}\\ & \Rightarrow I_3>I_2>I_1\end{array}$$