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If $\mathrm{I}$ is the greatest of $I_1=\int_0^1 e^{-x} \cos ^2 x d x, I_2=\int_0^1 e^{-x^2} \cos ^2 x d x, I_3=\int_0^1 e^{-x^2} d x, I_4=\int_0^1 e^{-x^2 / 2} d x$, then
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The correct answer is:
$\mathrm{I}=\mathrm{I}_4$
$\mathrm{e}^{-\mathrm{x}} \cdot \cos ^2 \mathrm{x} < \mathrm{e}^{-\mathrm{x}^2} \cos ^2 \mathrm{x} < \mathrm{e}^{-\mathrm{x}^2} < \mathrm{e}^{-\frac{\mathrm{x}^2}{2}}$ for $0 < \mathrm{x} < 1$
$\therefore$ By domination Law, $\mathrm{I}_4$ is maximum
$\therefore \mathrm{I}=\mathrm{I}_4$
$\therefore$ By domination Law, $\mathrm{I}_4$ is maximum
$\therefore \mathrm{I}=\mathrm{I}_4$
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