Here are three very important facts about integrability:

[1]: If f is continuous on [a,b], then f is integrable on [a,b]

[2]: If f is bounded on [a,b] and has finitely many points of discontinuity on [a,b], then f is integrable on [a,b]

[3]: If f is not bounded on [a,b], then f is not integrable on [a,b].

Is your f continuous?. x+1 is just a line with y intercept at y=1 and slope 1.

There are no points of discontinuity on the line.

If f is defined on a closed interval [a,b], then it is Riemann integrable on [a,b] if the limit exists:

If it is integrable over [a,b] then we write:

Can you integrate f?. You have: