Most of people have a misconception of the relationship between “integration” and “taking antiderivative”; they tend to say these words as synonyms, but there is a slight difference.

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In general, “Integral” is a function associate with the original function, which is defined by a limiting process. Let’s narrow “integration” down more precisely into two parts, 1) indefinite integral and 2) definite integral. Indefinite integral means integrating a function without any limit but in definite integral there are upper and lower limits, in the other words we called that the interval of integration.

While an antiderivative just means that to find the functions whom derivative will be our original function. There is a very small difference in between definite integral and antiderivative, but there is clearly a big difference in between indefinite integral and antiderivative. Let’s consider an example:

f(x) = x²

The antiderivative of x² is F(x) = ⅓ x³.

The indefinite integral is ∫ x² dx = F(x) = ⅓ x³ + C, which is almost the antiderivative except c. (where “C” is a constant number.)

On the other hand, we learned about the Fundamental Theorem of Calculus couple weeks ago, where we need to apply the second part of this theorem in to a “definite integral”.

The definite integral, however, is ∫ x² dx from a to b = F(b) – F(a) = ⅓ (b³ – a³).

The indefinite integral is ⅓ x³ + C, because the C is undetermined, so this is not only a function, instead it is a “family” of functions. Deeply thinking an antiderivative of f(x) is just any function whose derivative is f(x). For example, an antiderivative of x^3 is x^4/4, but x^4/4 + 2 is also one of an antiderivative. Despite, when we take an indefinite integral, we are in reality finding “all” the possible antiderivatives at once (as different values of C gives different antiderivatives). So there is subtle difference between them but they clearly are two different things. In additionally, we would say that a definite integral is a number which we could apply the second part of the Fundamental Theorem of Calculus; but an antiderivative is a function which we could apply the first part of the Fundamental Theorem of Calculus.

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This entry was posted in Uncategorized on January 25, 2017 by moiz ali.

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5 thoughts on “Integral vs Antiderivative”

Sean Manoukian April 10, 2021 at 11:04 am

Thanks for this, it’s very helpful. But I am wondering if there is a typo in the final paragraph, here:

“For example, an antiderivative of x^3 is x^4/4”

Shouldn’t that be 1/4 x^4 instead of x^4/4?

Anyway, thanks!

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