It is not just the preposition to that takes the gerund, but the construction "be adjective to":
Locking him up without food or water is tantamount to killing him.
Knowing your budget is important to successfully planning your holiday.
There are lists of verbs that take an infinitive or a gerund, like here, but they are seldom complete.
The gerund is also after verbs like "looking forward to":
I look forward to meeting you
He looks forward to getting to know her better.
Let's have a look at another example sentence to see if there is a difference between "be adjective to gerund" and "be _adjective infinitive":
Sharp knifes are important to cooking.
Sharp knifes are important to cook.
In the first sentence, I state that an important part of the action of cooking consists of sharp knifes. The sharp knifes can be seen as a property of cooking.
The second sentence is different. Well, there is the possible confusion that I mean I should cook my knifes, for one. That makes little sense.
Even if we ignore that, I would parse the sentence more like:
If I am to cook, it is important I have sharp knifes.
So if I have no sharp knives, I won't start cooking.
Somehow, the sentence gives me more of a specific feeling, whereas the cooking variant is a general observation.
Now, in your original sentence:
changes are integral to achieving improvements.
changes are integral to achieve improvements.
The first sentence makes changes an (integral) property of "achieving improvements" in general.
The second one seems to presuppose we have a specific situation where we want to achieve improvements, and in order to do that, we need changes.