12th Class
Mathematics
Definite Integrals
Question Bank
Critical Thinking
question_answer
If \[I\] is the greatest of the definite integrals \[{{I}_{1}}=\int_{0}^{1}{{{e}^{-x}}{{\cos }^{2}}x\,dx},\] \[{{I}_{2}}=\int_{0}^{1}{{{e}^{-{{x}^{2}}}}}{{\cos }^{2}}x\,dx\] \[{{I}_{3}}=\int_{0}^{1}{{{e}^{-{{x}^{2}}}}dx},\] \[{{I}_{4}}=\int_{0}^{1}{{{e}^{-{{x}^{2}}/2}}dx},\] Then
A)\[I={{I}_{1}}\]
B)\[I={{I}_{2}}\]
C)\[I={{I}_{3}}\]
D)\[I={{I}_{4}}\]
Correct Answer:
D
Solution :
For\[0<x<1\], we have \[\frac{1}{2}{{x}^{2}}<{{x}^{2}}<x\]
\[\Rightarrow -{{x}^{2}}>-x,\]so that \[{{e}^{-{{x}^{2}}}}<{{e}^{-x}}\],