How to Integrate x²? (Formula & Solution)

To solve this, you do not need to do any massive, complex math. You simply use the universal Power Rule of Integration. The Power Rule states that if you want to integrate any standard variable 'x' raised to a power 'n' (as long as n is not -1), the formula is: ∫ xⁿ dx = [ xⁿ⁺¹ / (n + 1) ] + C

In simple English: You take the old power, ADD exactly 1 to it, and then divide the entire thing by that brand new, bigger power.

Let us apply the Power Rule to our specific problem (∫ x² dx):

1. Identify the power 'n'. Here, our n = 2.
2. Add 1 to the power: 2 + 1 = 3.
3. Make the new power the exponent: x³.
4. Divide the whole expression by the new power (3): (x³ / 3).
5. Finally, ALWAYS add the massive letter 'C' at the end.

Final Result: (x³ / 3) + C

The 'C' stands for the Constant of Integration. Integration is basically just the mathematical reverse of Differentiation. If we differentiate the function [ (x³/3) + 5 ], the answer is x². If we differentiate [ (x³/3) - 99 ], the answer is also x² (because the derivative of any plain number is 0). Because the derivative destroyed the original secret number, when we reverse the process (integrate), we do not know what that hidden number was! So, we write a big '+ C' to mathematically represent that unknown, destroyed constant.