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If int(0)^(x)f(t)dt=e^(x)-ae^(2x)int(0)^...

If `int_(0)^(x)f(t)dt=e^(x)-ae^(2x)int_(0)^(1)f(t)e^(-t)dt`, then

A

`a=(1)/(3-2e)`

B

`f(x)=e^(x)-2e^(2x)`

C

`a=(1)/(e)`

D

`f(x)=e^(x)-e^(-x)`

Text Solution

Verified by Experts

The correct Answer is:
A, B
Put `x=0rArr e^(0)-a int_(0)^(1)f(t)e^(-t)dt=0`
`rArr" "int_(0)^(1)f(t)e^(-1)dt=(1)/(a)" (i)"`
`rArr" "int_(0)^(x)f(t)dt=e^(x)-ae^(2x)(1)/(a)=e^(x)-e^(2x)`
Differentiating we get
`rArr" "f(x)=e^(x)-2e^(2x)`
From (i), we get `a=(1)/(3-2e)`
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