I'm currently working on a question out of Otto Bretscher's "Linear Algebra with Applications," in the chapter regarding isomorphic transformations, and it's got me completely stumped.
I have to solve question 70 in this image
I started trying to express the integral itself as a linear transformation from $P_{n-1}$ to $R$, and I got the row transformation matrix
[2, 0, 2/3, 0, 2/5, 0, 2/7, 0, ... 2/n]
in the basis [1, t, t^2, t^3, ... t^{n-1}],
as it seems the rest of the section would lead me to do, but I don't really understand how to get to the "weights" mentioned in the problem. I feel like I just lack a fundamental understanding of what the problem is really asking, and any help would be appreciated!