First slide

Evaluation of definite integrals

Question

If I is the greatest of the definite integrals
$\begin{array}{ll} I_{1} = \int_{0}^{1} e^{- x} \cos^{2} x d x, I_{2} = \int_{0}^{1} e^{- x^{2}} \cos^{2} x d x, \\ I_{3} = \int_{0}^{1} e^{- x^{2}} d x, I_{4} = \int_{0}^{1} e^{- x^{2} / 2} d x \end{array}$
then

Moderate

A

$I = I_{1}$
B

$I = I_{2}$
C

$I = I_{3}$
D

$I = I_{4}$

Solution

For $0 < x < 1$ , we have $(1 / 2) x^{2} < x^{2} < x$ i.e., $- x^{2} > - x$ so that $e^{- x^{2}} > e^{- x}$
Hence $\int_{0}^{1} e^{- x^{2}} \cos^{2} x d x >$
$\int_{0}^{1} e^{- x} \cos^{2} x d x$ . Also $\cos^{2} x \leq 1$ , therefore
$\int_{0}^{1} e^{- x^{2}} \cos^{2} x d x \leq \int_{0}^{1} e^{- x^{2}} d x < \int_{0}^{1} e^{- x^{2} / 2} d x = I_{4} .$
Hence $I_{4}$ is the greatest integral.

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