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A function f, continuous on the positive...

A function f, continuous on the positive real axis, has the property that for all choices of x gt 0 and y gt 0, the integral `int_(x)^(xy)f(t)dt` is independent of x (and therefore depends only on y). If f(2) = 2, then `int_(1)^(e)f(t)dt` is equal to

A

e

B

4e

C

4

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
C
Since, I is independent of x, so `(dI)/(dx)=0`
`rArr" "f(xy).y-f(x)=0`
`rArr" "f(x.y).y=f(x)`
Put, `y=(2)/(x),f(2)((2)/(x))=f(x)`
`rArr" "f(x)=(4)/(x)`
`rArr" "int_(1)^(e)(4)/(x).dx=4 ln e = 4`
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