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I am learning this currently in calculus but I don't understand the actual difference logically. I can answer the questions but I don't know why the answer is what it is. Any help is greatly appreciated.

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  • $\begingroup$ One (I forget which is the first and which is second) is about differentiating an integral and the other is about integrating derivatives. Try taking a look at the Wikipedia article for more info. $\endgroup$
    – user137731
    Dec 1 '15 at 22:34
  • $\begingroup$ @Bye_World Usually the integral of the derivative is "second", probably because it is harder to prove. $\endgroup$
    – Ian
    Dec 1 '15 at 23:02
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As Bye_World has indicated, the two theorems are opposites of each other. The first theorem states that under suitable conditions on $f$ and an arbitrary $a$ where $f$ is integrable, $$\frac{d}{dx}\int_a^x f(x)\ dx = f(x)$$ That is, differentiation undoes integration. The second theorem states that under suitable conditions on $f$, $$\int_a^x \frac{df}{dt}(t)\ dt = f(x) - f(a)$$ That is, integration undoes differentiation (up to a constant).

In fact, if we were willing to put up with tighter restrictions on the function, we could easily prove either one from the other. But those restrictions are inconvenient, so instead we have the proofs you will find in your calculus book, so the conditions on $f$ are looser.

When you see the phrase "Fundamental Theorem of Calculus" without reference to a number, they always mean the second one. The first theorem is instead referred to as the "Differentiation Theorem" or something similar.

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