You can always do quadgk(...)[1] to get the first element of the tuple (the integral) without assigning it to a variable.
Note also that quadgk(u -> f(u), ...) is equivalent to quadgk(f, ...)
Finally note that, If you are calculating F(x) = quadgk(f, 0, x) for a whole bunch of different values of x ∈ [0,a], then there are much more efficient things that you can do than separate quadgk integrals for each x, which wastes a lot of calculations that could be shared between different x values, especially if f(x) is a smooth function. For example, using ApproxFun.jl, you can do F = cumsum(Fun(f, 0..a)), and then evaluate F(x) very quickly — this works by first constructing a polynomial approximation of f(x) on [0,a] and then forming the polynomial F(x) that is the indefinite integral.