TITLE: Ambiguous integration
QUESTION [5 upvotes]: 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?

REPLY [4 votes]: The change of variable $u=x^2$, $du=2xdx$ gives
$$\int_0^3 xf(x)\,dx={1\over2}\int_0^9f(u)\,du={10\over2}=5$$
The other piece of information, $\int_0^3f(u)\,du=14$, is irrelevant.  One wonders why it was mentioned.