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Evaluate: int0^oo[2e^(-e)]dx ,w h e r e[...

Evaluate: `int_0^oo[2e^(-e)]dx ,w h e r e[x]` represents greatest integer function.

Text Solution

Verified by Experts

The correct Answer is:
In2
`f(x)=2e^(-x)` is decreasing for `x epsilon [0,oo)`.
Also, when `x=0, 2e^(-x)=2`,
and when `x to oo,2e^(-x)to0`.
Thus `[2e^(-x)]` is discontinuous when `2e^(-x)=1` or `x=log2`
Also for `xgtIn2, [2e^(-x)]=0`
and for `0ltxltlog2, [2e^(-x)]=1`
`:.int_(0)^(oo) [2e^(-x)]dx=int_(0)^(In2)[2e^(-x)]dx+int_(In2)^(oo) [2e^(-x)]dx`
`=int_(0)^(In2) 1dx+int_(In2)^(0)0dx=(x)_(0)^(In2)=In2`
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