Ambiguous integration I had come across the following question:
If we know $\int_0^9 f(u)du = 10$ and $\int_0^3 f(u)du = 14$ then what must the value of $\int_0^3 xf(x^2)dx$ be?
I reason that this value would be $30$ because if $\int_0^3 f(u)du = \int_0^3 f(x)dx$ then $\int_0^3 f(x^2) dx$ then evaluating for $x$ would be the same as evaluating from $0$ to $9$. Therefore $\int_0^3 f(x^2)dx = \int_0^9 f(u)du = 10$. Then all we do is multiply the value of $f(x^2)$ by 3 when we are evaluating the entire integral. Thus we end up with the answer $30$.
Would this reasoning be correct?

Begin with
$$\int_0^3 xf(x^2) dx,$$
and perform the $u$-substitution $u = x^2$ Then $du = 2x dx$, so that
$$\frac{1}{2} \int_0^3 f(x^2) 2xdx = \frac{1}{2}\int_0^9 f(u) du = 5.$$
Note that in your question, you claimed something different than the first equality. You claimed that
$$ \color{#AA0000}{\int_0^3 f(x^2)dx = \int_0^9 f(u)du},$$
which is not true. (I put it in red to emphasize that it's not a true statement). 
Returning to the previous statement, this means that
$$\int_0^3 xf(x^2) dx = 5.$$
