A) \[{{I}_{2}}>{{I}_{3}}>{{I}_{1}}\]
B) \[{{I}_{3}}>{{I}_{1}}>{{I}_{2}}\]
C) \[{{I}_{2}}>{{I}_{1}}>{{I}_{3}}\]
D) \[{{I}_{3}}>{{I}_{2}}>{{I}_{1}}\]
Correct Answer: D
Solution :
Given: \[{{I}_{1}}=\int_{0}^{1}{{{e}^{-x}}{{\cos }^{2}}xdx;}\] \[{{I}_{2}}=\int_{0}^{1}{{{e}^{-{{x}^{2}}}}{{\cos }^{2}}xdx}\] and \[{{I}_{3}}=\int_{0}^{1}{{{e}^{-{{x}^{3}}}}dx}\] For \[x\in (0,1),x>{{x}^{2}}\Rightarrow -x<-{{x}^{2}}\] \[\Rightarrow {{x}^{2}}>{{x}^{3}}\] \[\Rightarrow -{{x}^{2}}<-{{x}^{3}}\] \[\Rightarrow {{e}^{-{{x}^{2}}}}<{{e}^{-{{x}^{3}}}}\] And \[{{e}^{-x}}<{{e}^{-{{x}^{2}}}}\] \[\Rightarrow {{e}^{-x}}<{{e}^{-{{x}^{2}}}}<{{e}^{-{{x}^{3}}}}\] \[\Rightarrow {{e}^{-{{x}^{3}}}}<{{e}^{-{{x}^{2}}}}>{{e}^{-x}}\] \[\Rightarrow {{I}_{3}}>{{I}_{2}}>{{I}_{1}}\] Green line denotes \[\int_{{}}^{{}}{(x)={{e}^{-x}}{{\cos }^{2}}x}\] Blue line denotes \[g(x)={{e}^{-{{x}^{3}}}}{{\cos }^{2}}x\] Red line denotes \[h(x)={{e}^{-{{x}^{3}}}}\] Also, from the graph we get the same result. Hence, option D is correct.You need to login to perform this action.
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