TITLE: Finding $\int_{-2}^8xf(x)dx$ given $\int_{-2}^8f(x)dx$
QUESTION [10 upvotes]: I have a continuous function $f:[-2,8]\rightarrow\mathbb{R}$ for which is true that $f(6-x)=f(x)\forall x\in[-2,8]$. Let:
$$\int_{-2}^8f(x)dx=10$$ Now, I want to find the: 
$$\int_{-2}^8xf(x)dx$$
I am thinking of using both the methods of u-substitution and integration by parts, but I need some help. Any ideas?

REPLY [26 votes]: Here's a cute trick.
If the problem is well-posed, then the solution must be independent of $f$. Therefore, you can take
$$
f(x)\equiv1
$$
which is consistent with the hypotheses, and calculate
$$
\int_{-2}^8x\ \mathrm dx\equiv 30
$$
Easy peasy!