Q.

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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a

I=I1

b

I=I2

c

I=I3

d

I=I4

answer is D.

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Detailed Solution

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.
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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