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So, I am aware of the differential calculus (derivatives) and the integral calculus (integrals).

And separate from that, there is the first fundamental theorem (FFT) of the calculus which relates the two processes as inverses of each other. So far, so good.

Now I would like to integrate, say, x-squared. HOWEVER, I would like to do it without the FFT.

I mean the following: yes, I know that (1/3)x-cubed is the answer (let's not quibble over constants or boundaries, or definite or indefinite). But I know that is the answer because when I take its derivative, I get x-squared. But that is using my knowledge of the FFT.

Can someone explain to me how to integrate x-squared without using the FFT? I am lost.

How did one do integrals BEFORE the FFT revealed it to be the inverse of differentiation?

Or am I suffering from OCD and barking up the wrong tree?