Definite integrals proof.

Question

I am reading this text and I don't get the proof:

I don't see how the FTC2 is applied a second time. Where did the second set of equations come from?

Where did the $u$ come from? And by "second time" do they mean "let's apply FTC2 to a completely different equation and show we get the same result?"

Answer 1

You look at $F(g(b))-F(g(a))$ in two different ways.

• $F(g(b))-F(g(a))=F(g(x))|_{a}^{b}$
• $F(g(b))-F(g(a))=F(u)|_{g(a)}^{g(b)}$