antiderivative

But the antiderivative we have here looks like this:

Wolfram Blog : Mathematica and the Fundamental Theorem of Calculus 2008

Differentiating this shows that it is indeed an antiderivative of the same function as above.

Wolfram Blog : Mathematica and the Fundamental Theorem of Calculus 2008

Because, when you state it fully, the theorem requires that the antiderivative that is going to be subtracted be continuous throughout the interval.

Wolfram Blog : Mathematica and the Fundamental Theorem of Calculus 2008

And indeed one can show that there is no single choice of antiderivative for which the Fundamental Theorem will always work.

Wolfram Blog : Mathematica and the Fundamental Theorem of Calculus 2008

Moreover, if a meromorphic integrand has simple poles in the complex plane, it is impossible to choose an antiderivative continuous along every imaginable path in the complex plane – because of branch cuts in.

Wolfram Blog : Mathematica and the Fundamental Theorem of Calculus 2008

The Fundamental Theorem of Calculus states that an antiderivative continuous along a chosen path always exists.

Wolfram Blog : Mathematica and the Fundamental Theorem of Calculus 2008

Goldman denied it, but the charge hardened antibank, anti-Wall Street, antiderivative sentiment in Congress.

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Another thing it can do is to make use of ambiguity in the antiderivative.

Wolfram Blog : Mathematica and the Fundamental Theorem of Calculus 2008

Sometimes it works by detecting discontinuities in the antiderivative, and then breaking up the integration region into parts, and carefully taking directional limits at the discontinuity points:

Wolfram Blog : Mathematica and the Fundamental Theorem of Calculus 2008

English - German absolute value - Betrag antiderivative - Stammfunktion approach ("limit of 1 over x as x approaches infinity") - annähern ("Grenzwert von 1 durch x für x gegen unendlich")

Statistical Terms in German C N Heidelberg 2009