Using the fundamental theorem of Calculus

Question

I have a couple of questions. I have them answered mostly I think, but I haven't done these before so hoping for some confirmation that I am right.

The question is to use the fundamental thm of calculus to find $F'(x)$ if

$(1) F(x) = \int_{0}^{x} \sqrt{1+t^2} dt$

To which my answer is $F'(x) = \sqrt{1+x^2}$ ?

$(2) F(x) = \int_{1}^{x} \frac{dt}{t} dt.$

Is this question written wrong in the notes, as it has two dts?

$(3) \int_{1}^{2x}\cos(t^2) dt.$ I used the chain rule and got $2\cos(4x^2) ?$

Thank you for any help provided.

Answer 1

You are right in every case. In particular, yes, the double $\mathrm dt$ is a mistake.

Answer 2

(1) Your answer is right.

(2) Yes, the question is written wrong.

(3) Your answer is right.