Assertion (A) : ∫01e−xcos2xdx<∫01e−x2cos2x dxReason (R): ∫abf(x)dx<∫abg(x)dx ∀ f(x)≥g(x)

0<x<1 −x<−x2e−x<ex2 ∫01e−xcos2xdx<∫01e−x2cos2xdx and if f(x)≥g(x) ∫abf(x)dx≥∫abg(x)dx

Assertion (A) : ∫01e−xcos2xdx<∫01e−x2cos2x dxReason (R): ∫abf(x)dx<∫abg(x)dx ∀ f(x)≥g(x)

0<x<1 −x<−x2e−x<ex2 ∫01e−xcos2xdx<∫01e−x2cos2xdx and if f(x)≥g(x) ∫abf(x)dx≥∫abg(x)dx

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Assertion (A) : $\int_{0}^{1} e^{- x} \cos^{2} x d x < \int_{0}^{1} e^{- x^{2}} \cos^{2} x d x$
Reason (R): $\int_{a}^{b} f (x) d x < \int_{a}^{b} g (x) d x \forall f (x) \geq g (x)$

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Statement-I is true; statement –II is true; statement-II is a not a correct explanation for statement-I

Statement-I is false; statement-II is true

Statement-I is true; statement –II is true; statement-II is a correct explanation for statement-I

Statement-I is true; statement-II false

answer is C.

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Detailed Solution

$0 < x < 1 \begin{array}{l} - x < - x^{2} \\ e^{- x} < e^{x^{2}} \end{array} \begin{array}{l} \int_{0}^{1} e^{- x} \cos^{2} x d x < \int_{0}^{1} e^{- x^{2}} \cos^{2} x d x and \\ if f (x) \geq g (x) \end{array} \int_{a}^{b} f (x) d x \geq \int_{a}^{b} g (x) d x$