Solving heat equation (BVP, PDEs)

[frozen totem]

The first image has the problem at the top. It asks to solve the given heat equations with given conditions. I need help with calculating the general u(x,t). As you can see at the bottom of the first image, there is a term in my answer that isn't in the memo. I have attached the full memo in the second image.

I would very much appreciate some assistance with spotting where my mistake is in my workings please. I have written out what I need help with in purple on the bottom of the first image.

[frozen totem]

It is very late here on my side of the world, so just keep in mind, if you are so kind as to respond, I myself will only be able to respond later tomorrow morning. Thank you in advance!

[uneven bloom]

You forgot the minus sign in front of the 2nd integral

[frozen totem]

Aa thank you

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