MECH 360 – Question about strain energy problem with beams

1. The problem statement, all variables and given/known data
I need to find the total strain energy in the beam with a moment applied:

3. The attempt at a solution
For my attempt I decided to split the integral into two parts since the moment is different depending on whether or not you are on the left side of the moment or on the right side.

I know there is an easier method; that the loading has a kind of skew-symmetry, so you might be able to take a short cut by evaluating the left side integral with a coordinate system x starting from A and going to the right, and the right side integral with a coordinate system X starting from B and going to the left.

However, for my attempt I decided to do the integral I circled with red below. When I run through the algebra I do not get the correct answer which is:

So I would like to know… Is my method correct or did I just do my algebra wrong. I feel like what I did should work… I spent a few hours on this and kept getting the wrong answer (might be just because I was tired and slept very little :zzz:)

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