Q94 / 155 · WBJEE 2022 · Mathematics

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

  • A
    $\mathrm{I}=\mathrm{I}_1$
  • B
    $\mathrm{I}=\mathrm{I}_2$
  • C
    $I=I_3$
  • Correct answer
    D
    $\mathrm{I}=\mathrm{I}_4$

Solution

$\mathrm{e}^{-\mathrm{x}} \cdot \cos ^2 \mathrm{x} < \mathrm{e}^{-\mathrm{x}^2} \cos ^2 \mathrm{x} < \mathrm{e}^{-\mathrm{x}^2} < \mathrm{e}^{-\frac{\mathrm{x}^2}{2}}$ for $0 < \mathrm{x} < 1$ $\therefore$ By domination Law, $\mathrm{I}_4$ is maximum $\therefore \mathrm{I}=\mathrm{I}_4$

Hence, the correct answer is option: 'D'.

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