A) \[{{I}_{2}}>{{I}_{1}}\]
B) \[{{I}_{1}}>{{I}_{2}}\]
C) \[{{I}_{3}}={{I}_{4}}\]
D) \[{{I}_{3}}>{{I}_{4}}\]
Correct Answer: B
Solution :
\[{{I}_{1}}=\,\,\int\limits_{0}^{1}{{{2}^{{{x}^{2}}}}dx,\,\,{{I}_{2}}=\int\limits_{0}^{1}{{{2}^{{{x}^{3}}}}dx,\,{{I}_{3}}=}\int\limits_{1}^{2}{{{2}^{{{x}^{2}}}}dx,\,{{I}_{4}}=}\int\limits_{1}^{2}{{{2}^{{{x}^{3}}}}dx}}\] \[\forall \,0<x<1,\,\,{{x}^{2}}>{{x}^{3}}\] \[\Rightarrow \,\,\,\int\limits_{0}^{1}{{{2}^{{{x}^{2}}}}dx,\,\,>\int\limits_{0}^{1}{{{2}^{{{x}^{3}}}}dx\,\,\Rightarrow \,}\,{{I}_{1}}>{{I}_{2}}}\] Also\[\forall \,1<x<2\,\,\,{{x}^{2}}<{{x}^{3}}\,\,\,\Rightarrow \int\limits_{1}^{2}{{{2}^{{{x}^{2}}}}dx<\int\limits_{1}^{2}{{{2}^{{{x}^{3}}}}dx\Rightarrow }\,\,{{I}_{3}}<{{I}_{4}}}\]You need to login to perform this action.
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