Answer
$f(t) = t^{3/2}$
$a = 9$
Work Step by Step
Consider the function $f(t) = t^{3/2}$
$6+\int_{a}^{x}\frac{f(t)}{t^2}~dt$
$= 6+\int_{a}^{x}\frac{t^{3/2}}{t^2}~dt$
$= 6+\int_{a}^{x}t^{-1/2}~dt$
$= 6+2t^{1/2}~\vert_{a}^{x}$
$= 6+2\sqrt{x}- 2\sqrt{a}$
To find the required value of $a$, we can set this expression equal to $2\sqrt{x}$:
$6+2\sqrt{x}- 2\sqrt{a} = 2\sqrt{x}$
$2\sqrt{a} = 6$
$\sqrt{a} = 3$
$a = 9$
Therefore:
$f(t) = t^{3/2}$
$a = 9$