How do I differentiate and integrate powers of x?

For differentiation, we can follow a simple method for differentiating y with respect to x. It may be the case that you have special numbers such as e^x you need to differentiate and integrate but these formulas are easily findable online.
The general rule of differentiation is that for y = ax^b, dy/dx = abx^b-1 , so for example if we had y = 3x² then dy/dx = 6x. We have multiplied the 3 by 2 and subtracted the original 2 in order to complete the differentiation.
For integration, you would do the reverse, so that for dy/dx = ax^b you would integrate it to get y = (ax^b+1)/b+1 + c, and we can easily check this. But we need to ask, where did that constant come from? This is easy, as for any constant c, c = c * 1 = cx⁰ as any number to the power of 0 is 1! So when the 0 carries over to the left hand side it removes the whole number. So just to check, by differentiating the c we remove it, we times our fraction by b+1 removing the demominator, and then b+1-1 = b which gets us back to ax^b! So for example, if we had dy/dx = 3x²+ 2x + 2, we would end up with y = (3x³)/3 + (2x²)/2 + (2x¹)/1 + c leading to y = x³ + x² + 2x + c! If given x and y values where this holds we can quickly work out c. If we wanted to find the area between two x values on a curve we would substitute them in here.