Are you looking for something like plot(x -> quadgk(f, 1, x)[1])?
That is, if you want to take advantage of the fact that the first plot argument can be a function, you pass a function x -> quadgk(f, 1, x)[1] that takes an x and returns \int_1^x f(x')dx', using [1] to get the first return value of quadgk (which returns a tuple).