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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!

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