I am a programmer and I am working on a website. I want to find out how to integrate a blockchain into my website. (Please note that the wallet address changes as soon as it receives a payment). I'm particular about how to resolve this issue.
You can install your own wallet and use it ie. with RPC
Probably you will need some payment processor. There are several, just google for them and their developers guides.
Many shop solutions like ie. magento have several extensions using different technology, you can look up how they do it and build based on their implementation.
Here is one of many guides: https://code.tutsplus.com/tutorials/adding-bitcoin-payment-processing-to-your-website--net-33098
First, I suggest that you learn a little more before trying to write software that utilizes bitcoin. You're clearly not informed about many things.
For instance, your wallet address doesn't change when it receives a payment. An additional address is generated, but the old address is still there and still can accept payments.
This is such a fundamental lack of understanding that it indicates you have almost 0 experience with bitcoin, which is why I suggest further research on your part.
You also need to understand exactly what "integrate" means to you, before you can write software to do it.