Home
Class 12
MATHS
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`
Promotional Banner

Similar Questions

Explore conceptually related problems

If f(-x)+f(x)=0 then int_a^x f(t) dt is

lim_(x to 0)(int_(-x)^(x) f(t)dt)/(int_(0)^(2x) f(t+4)dt) is equal to

If f(x) =int_(0)^(x) sin^(4)t dt , then f(x+2pi) is equal to

Let f ( x) = int _ 0 ^ x g (t) dt , where g is a non - zero even function . If f(x + 5 ) = g (x) , then int _ 0 ^ x f (t ) dt equals :

If f(x) = int_(0)^(x)(2cos^(2)3t+3sin^(2)3t)dt , f(x+pi) is equal to :

If" f, is a continuous function with int_0^x f(t) dt->oo as |x|->oo then show that every line y = mx intersects the curve y^2 + int_0^x f(t) dt = 2

If" f, is a continuous function with int_0^x f(t) dt->oo as |x|->oo then show that every line y = mx intersects the curve y^2 + int_0^x f(t) dt = 2

If f(x)=int_(x^2)^(x^2+1)e^(-t^2)dt , then f(x) increases in

If int_(0)^(x)((t^(3)+t)dt)/((1+3t^(2)))=f(x) , then int_(0)^(1)f^(')(x) dx is equal to

If int_(0) ^(x) f (t) dt = x + int _(x ) ^(1) t f (t) dt, then the value of f (1) , is