## Fundamental theorem of calculus example

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:

.…