Integral representations of the Fibonacci numbers

Question

I would like to assemble a thorough list of integral representations of various number sequences, and the Fibonacci numbers are naturally my first choice. So, my question:

What are some integral representations of the Fibonacci numbers?

And a related question:

Typically, we let $F_1,F_2 = 1$. Let $F_n(a,b)$ denote the $n$-th Fibon-$ab$ number, with $F_1 = a$, $F_2= b$, and a recursion rule identical to that of the normal Fibonacci numbers. What integral representations are known for Fibon-$ab$ numbers for certain pairs (or broader classes of pairs) $a,b$?

I will leave one answer to get this post rolling.

Answer 1

This formula comes from this Wolfram page:

$$F_{2n}= \frac n2 \left(\frac32\right)^{n-1} \int_0^ \pi \left(1+\frac{\sqrt 5}{3}\cos (x) \right)^{n-1} \sin (x)~dx.$$

Answer 2

$$F_n= \frac{n}{\sqrt{5}} \int_{b}^{a} x^{n-1} dx, a,b =\frac{1\pm \sqrt{5}}{2}$$