We did an example today in class which I wanted to go through again here. The question was to calculate

**We spot the pattern immediately that it’s an FTC part 1 type question**, but it’s not quite there yet. In the FTC part 1, the upper limit of the integral is just , and not . A question that we would be able to answer is:

This would just be . Or, of course, we can show that **in exactly the same way:**

That’s just changing the names of the variables, which is fine, right? But that’s not quite the question. So, **how can we convert from to **? Well, how about a substitution? How about letting and seeing what happens. This is actually just a chain rule. It’s like if I asked you to calculate:

.

You would just say: Let and then we have:

.

Doing EXACTLY the same thing in this case gives us:

.

This now allows us to solve any such question. **Have a look at the questions in this week’s tutorial** and see if you can get them. Note that one other piece of information which you know, but you will need to remember is that:

.

Let me know in the comments if this post has been helpful.

**Conceptually**, what is really happening here is we are saying: How quickly does the area under the curve of change when we change the upper limit, but the position of the upper limit is changing as as we vary .

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