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Symbolic solve

Following the change of variable thread, I wanted to streamline the whole process.

Namely, using the same example in the above thread, I'd like to say

integral_def_change(x*cos(x^2+1), (x, 0, 2*pi), u==x^2+1, u)

The difference is, I wanted also Sage to automatically solve for x instead of providing x= sqrt(u-1), say. But when I tried

solve(u==x^2+1, x)[0].rhs()

the output was r1 ???

1- What exactly is r1 ??

A way out (see this thread) seems to make of the solution a function of u

f(u)=solve(u==x^2+1, x)[0].rhs()

Now f is

u |--> -sqrt(u - 1)

2- What can I do to get +sqrt(u - 1) instead? Is this related to the positive function question there?