How to find $a$ that solves the following equation for $x > 0$?

Question

Can someone please help me figure out how to solve this? $$ 6 + \int\limits_a^{x^2}\frac{f(t)}{t^2}dt = 2\sqrt{x}. $$

Answer 1

Note that if $x^2=a$, then the integral is zero. Therefore, if $x=\sqrt{a}$, then

$$ 6=2\sqrt{\sqrt{a}}. $$

Can you take it from here?

Note: If you wanted to find $f$, you can take the derivative of both sides to get

$$ \frac{f(x^2)}{x^4}(2x)=\frac{1}{\sqrt{x}}. $$ Then, solve for $f(x^2)$ and substitute $y=x^2$. This will give you a formula for $f$.