A calculus problem by Utkarsh Singh

Now, given that

$f(x)\quad =\quad f(2x)\quad =\quad f(4x)...$

we can write it as

$f(x)\quad =\quad f({ 2 }^{ n }x)$

replace x by ${ 2 }^{ n }$ to get

$f(x)\quad =\quad f({ (1/2 })^{ n }x)$

Now put $lim\\ n->\infty$ to obtain

$f(x)\quad =\quad f(0)\quad =\quad 0\quad (given)$

This means that f(x) is a constant function whose value is zero. Now put the value of f(x) in the integral to obtain the given asnwer!