Write the integral of-------3---------x----squared----------- with respect to x as
-------------------∫-----------------3---------x---------2---------------------------------dx---------------and "compress":
--------------------------------------∫3x2dx------------------------------------.
Inspection of 3x2 shows this to be a simple integration.
In this case, the procedure involves "stepping-up" the exponent of x in 3x2 from 2
to 3. Writing to reflect this step would give an incomplete answer of 3x(2+1) or 3x3.
Now consider the coefficient to the left of the x in 3x3, which is a 3.
This coefficient of 3 in 3x3 needs to be changed in order to
give a product of 3 when the exponent of x (also a 3) "flies"
over the x during differentiation, leaving
an exponent of 2 behind it and then multiplying by the
coefficient already to the left of x to give the coefficient of
3 in 3x2.
This is accomplished by changing 3x3 in the incomplete answer to 1x3 or x3.
As a complete answer, Mathematicians write ∫3x2dx = x3 + C, where C is a "Constant".
As a check, one takes the derivative of x3 + C, following the rule d(xn)/dx = nx(n-1).
Then x3 would differentiate to 3x(3-1) or 3x2. C (being some exact number) would differentiate to 0.
Finally, this check returns 3x2 + 0 or 3x2, the "Integrand" given.
