Question
If I is the greatest of the definite integrals
I1=∫10e−xcos2x dx, I2=∫10e−x2cos2 x dx
I3=∫10e−x2dx, I4=∫10e−x22dx, then
I1=∫10e−xcos2x dx, I2=∫10e−x2cos2 x dx
I3=∫10e−x2dx, I4=∫10e−x22dx, then
Answer: Option D
:
D
For 0 < x < 1, we have 12x2<x2<x
⇒−x2>−x,sothate−x2<e−x,
Hence∫10e−x2cos2xdx>∫10e−xcos2xdx
Alsocos2x≤1
Therefore∫10e−x2cos2xdx≤∫10e−x2dx<∫10e−x22dx=I4
Hence I4 is the greatest integral
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:
D
For 0 < x < 1, we have 12x2<x2<x
⇒−x2>−x,sothate−x2<e−x,
Hence∫10e−x2cos2xdx>∫10e−xcos2xdx
Alsocos2x≤1
Therefore∫10e−x2cos2xdx≤∫10e−x2dx<∫10e−x22dx=I4
Hence I4 is the greatest integral
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