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A function f(x) satisfie f(x)=f((c)/(x))...

A function f(x) satisfie `f(x)=f((c)/(x))` for some real number c( gt 1) and all positive number 'x'. If `int_(1)^(sqrtc)(f(x))/(x)dx=3`, then `int_(1)^(c)(f(x))/(x)dx` is

A

4

B

6

C

8

D

9

Text Solution

Verified by Experts

The correct Answer is:
B
`I=int_(1)^(c)(f(x))/(x)dx=int_(1)^(sqrtc)(f(x))/(x)dx+int_(sqrtc)^(c)(f(x))/(x)dx`
Put `x=(c)/(t)` in second integral
`rArr" "dx=-(c)/(t^(2))dt`
`therefore" "I=int_(1)^(sqrtc)(f(x))/(x)dx+int_(sqrtc)^(1)(f((c)/(t)))/(t)(-dt)`
`" "=int_(1)^(sqrtc)(f(x))/(x)dx+int_(1)^(sqrtc)(f(x))/(x)dx=6`
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