The question is: The function f is differentiable on (−∞, ∞). Determine f (2) in the following cases:

$\displaystyle \int_0^{f(x)}{t^2}dx={x^2}{(1+x)}} $

So I let $\displaystyle {u}={f(x)}$ ==>$\displaystyle \int_0^u{t^2}{dt}$ ==> $\displaystyle \frac{d}{du} \int_0^u{t^2}{dt}={(f(u))^2}$

So $\displaystyle \frac{d}{dx} \int_0^u {t^2}{dt}={(f(u))^2} \frac{du}{dx}$ ==> $\displaystyle {(f(x))^2}{f'(x)}={2x}+{3x^2}$

Assuming that my steps are correct (and I'm not certain that they are), where do I go from here? For instance, what is $\displaystyle {f'(x)}$?

Many thanks for any suggestions for this problem!

CP