Why would you just take an integral? What reasoning are you using to think that would give you the range? I'm assuming by range you mean the min and max values of h(x) on this interval.

So for any closed interval, the min/max is going to occur either at relative extrema or at the endpoints. You need to take the derivative of h(x) and find the critical points, then compare them with the values of h(x) at the endpoints of the interval.