Integration variable : The next topic which we have to discuss here is the integration variable utilized in the integral. In fact there isn't actually a lot to discuss here other than to note as well that the integration variable doesn't actually matter. For instance,
∫x4 + 3x - 9 dx = (1/5) x5 + 3/2 x2 - 9x + c
∫t 4 +3t - 9 dt = 1 t 5 + 3 t 2 - 9t + c
∫w4 + 3w - 9 dw = 1 w5 + 3 w2 - 9w + c
Changing the integration variable in the integral merely changes the variable in the answer. It is significant to notice though that while we change the integration variable in the integral we also changed the differential (dx, dt, or dw) to match the new variable. it is more significant that we might realize at this point.
Another utilization of the differential at the ending of integral is to tell us what variable we are integrating with respect to. At this stage which may seem insignificant since mostly integrals which we're going to be working with here will only include a single variable. Though, if you are on a degree track which will take you into multi-variable calculus it will be very significant at that stage as there will be more than one variable in the problem. You have to get into the habit of writing the accurate differential at the end of the integral so while it becomes significant in those classes already you will be in the habit of writing it down.
To see why it is important take a look at the following two integrals.
∫ 2x dx ∫ 2t dx
The first integral is simple enough.
∫ 2x dx = x2 + c
The second integral is also rather simple, but we have to be careful. The dx tells us that we are integrating x's. That means that we integrate x's only which are in the integrand and all other variables in the integrand are assumed to be constants. Then the second integral is,
∫ 2t dx = 2tx + c
Thus, it may appear silly to always put in the dx, however it is a vital bit of notation which can cause us to acquire the incorrect answer if we neglect to put it in.