- #1

Trying2Learn

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- TL;DR Summary
- How can you prove the integral without knowing the derivative

Hello

(A continued best wishes to all, in these challenging times and a repeated 'thank you' for this site.)

OK, I have read that Newton figured out that differentiation and integration are opposites of each other.

(This is not the core of my question, so if that is wrong, please let it go.)

I can work out the derivative of, say, f(x) = x-squared.

However, I am having trouble proving the integral of, say, x-cubed.

If it is true, that it was 'discovered' that differentiation is anti-integration or integration is anti-derivative...

If that is true that it was 'discovered' and that people were studying differentiation and integration SEPARATELY, then I should be

able to derive the integral of say, x-cubed.

But I cannot do it without the first fundamental theorem of the calculus (which, assumes already that the two are opposites).

First, is this a ridiculous question?

If not, could someone work out one example? Something simple like the indefinite integral of say: x-cubed divided by 3

Thank you

(A continued best wishes to all, in these challenging times and a repeated 'thank you' for this site.)

OK, I have read that Newton figured out that differentiation and integration are opposites of each other.

(This is not the core of my question, so if that is wrong, please let it go.)

I can work out the derivative of, say, f(x) = x-squared.

However, I am having trouble proving the integral of, say, x-cubed.

If it is true, that it was 'discovered' that differentiation is anti-integration or integration is anti-derivative...

If that is true that it was 'discovered' and that people were studying differentiation and integration SEPARATELY, then I should be

able to derive the integral of say, x-cubed.

But I cannot do it without the first fundamental theorem of the calculus (which, assumes already that the two are opposites).

First, is this a ridiculous question?

If not, could someone work out one example? Something simple like the indefinite integral of say: x-cubed divided by 3

Thank you