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

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

Infinity Learn · Accepted Answer

For $$0$$−x so that e−$$x2$$>e−xHence ∫$$01$$ e−$$x2cos2$$⁡xdx>∫$$01$$ e−$$xcos2$$⁡xdx. Also  $$cos2$$⁡x≤$$1$$, therefore∫$$01$$ e−$$x2cos2$$⁡xdx≤∫$$01$$ e−$$x2dx$$<∫$$01$$ e−$$x2/2dx=I4$$.Hence $$I4$$ is the greatest integral.

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

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If I is the greatest of the definite integrals I1=∫01 e−xcos2⁡xdx, I2=∫01 e−x2cos2⁡xdx,I3=∫01 e−x2dx, I4=∫01 e−x2/2dxthen

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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