If f(x) and f(t) both have the same domain and range, is there a general way to find
$\int_{0}^{x^2} f(t) dt = f(x)$ given t? The actual problem tells that t = 9 and f(x) = $5 e \exp{x cos (x)cos(\frac{x}{sin x}) }$. A general way would be preferred, as that way I would be able to solve other problems. An example with a simpler f(x) would be appreciated. Also, the $x^2$ is just an arbitrary expression; it could be anything.
Thanks so much for your help!
I would like to give different names to the functions, say $f(x)$ and $g(t)$. I guess, what you want to ask is: What is $g(t)$ for given $t$ and $f(x)$?
First of all, your equation only gives information about $g(t)$ for $t>0$, so I will also assume $x>0$.
The answer can be found by taking the derivative of your integral equation with respect to $x$ using Leibniz' rule, which gives
$$f'(x)=\frac{d}{dx}\int_0^{x^2}dt g(t)=2xg(x^2). $$
By substituting $x=\sqrt{t}$, we get the result
$$g(t)=\frac{f'(\sqrt{t})}{2\sqrt{t}}.$$
Note, that this can also be written as $$g(t)=\frac{d}{dt}f(\sqrt{t}).$$