Integral problem w/ both limits as variable

Hey all, I'm very new to these forums. I'm not looking for the answers, but rather to figure out if the way I go about doing this problem is correct or if I am totally wrong any explanation would be very appreciated.

Question: Use the fundamental theorem of Calculus to find the derivative of the given function

What I did:

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**Ive never used LaTeX before so I hope this is the right way of showing it.

**If I messed up the whole thing or any part please give me tips on where I went wrong, thank you!

P.S - is there any online program that I could input an integral such as this so I can check myself?

Sorry for all the questions + thank you!