Cost of changing a battery for iPhone 5S?

My iPhone 5S battery is not doing so well. I've been told after a check that it will probably have to be changed.
In the store I got it checked at they told me the price of changing the battery would be about 120 pounds, but that they often instead of fixing the battery would "swap" the phone for a brand new one, which would then cost about 250 pounds.
After a quick internet search I found that many places offer to change the battery for about 40 pounds.
Is the store I went to just overcharging or are these cheaper places not to be trusted?

Why not replace it yourself? I have not worked on the iPhone until iPhone 4 and replace the battery is easy. There are plenty of videos on the Net will show you how. Specially on Youtube.com

This is where I bought my battery at:

https://www.ifixit.com/...F124-002-2

The video:

Hope it help.

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