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I am new to Monero, so I am just learning about Integrated Addresses vs. Standard Addresses.

I understand the difference between the two formats (an Integrated Address incorporates a Payment ID, which enables the recipient to determine who sent the funds).

I also understand that an Integrated Address can be generated algorithmically by combining a standard address with a payment ID.

However, it seems a common problem that I've run into is that various online services provide you with an Integrated Address, but certain cryptocurrency exchange platforms (such as Changelly) only accept Standard Address (at least for now).

So, my assumption is that it is safe to simply use a tool to programmatically derive the standard address from an integrated address. I use https://xmr.llcoins.net/addresstests.html and paste the Integrated Address into Box 8 and click "Check Address".

Then it generates a Standard Address in Box 15. I am assuming it is safe to transfer funds to the generated Standard Address, and the new funds will then show up in the wallet associated with the Integrated Address.

Again, I assume this, but am not sure. Since making a mistake here will cost time and money, I am asking this question to make sure my understanding is correct.

Can I simply send funds to the Standard Address (generated in Box 15), and expect those funds to appear in the wallet associated with my Integrated Address?

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As you have learnt, an integrated address is just a standard address with a payment ID, and yes, you can use tools like xmr.llcoins.net to extract the standard address (box 15) and payment ID (box 10.3) from an integrated address entered in box 8.

Can I simply send funds to the Standard Address (generated in Box 15), and expect those funds to appear in the wallet associated with my Integrated Address?

This depends. If the wallet associated with your integrated address is an exchange wallet (or any other type of 3rd party custodial wallet), this means the exchange expects the payment ID. Therefore, it is fine to send to the standard address so long as you also provide the payment ID. Almost all wallets provide the option to specify a standard address and separate payment ID. However, in this example, sending to the standard address without supplying the payment ID, is probably not a good idea, as the exchange clearly needs the payment ID for however they have implemented their accounting.

This is why there is a clear warning on that tool:

Note: box 15 is for informational purposes only! Do NOT use the address in 15 unless you know what you're doing!!

In any case, you should think long and hard about what exchanges or services you are using when they are custodial wallets. The phrase "Not your keys, not your coins", comes to mind. If you have the wallet keys, you can safely receive funds to the standard address without a payment ID.

I also understand that an Integrated Address can be generated algorithmically by combining a standard address with a payment ID.

Correct. See this answer.

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