Most of human being have a ofradiation of the relationship between “integration” and “taking antiderivative”; they tend to say these words together synonyms, however there is a slim difference.

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In general, “Integral” is a role associate through the original function, i beg your pardon is defined by a limiting process. Let’s small “integration” down more precisely right into two parts, 1) indefinite integral and 2) definite integral. Indefinite integral way integrating a role without any kind of limit yet in definite integral there are upper and lower limits, in the other words we dubbed that the term of integration.

While one antiderivative just method that to discover the attributes whom derivative will be our original function. There is a very little difference in between definite integral and antiderivative, but there is plainly a big difference in between indefinite integral and antiderivative. Let’s take into consideration 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 virtually the antiderivative except c. (where “C” is a consistent number.)

On the various other hand, we learned about the an essential Theorem of Calculus pair weeks ago, where we need to apply the second part of this to organize in to a “definite integral”.

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

The indefinite integral is ⅓ x³ + C, due to the fact that the C is undetermined, therefore this is not just a function, instead it is a “family” that functions. Deeply reasoning an antiderivative of f(x) is just any role whose derivative is f(x). Because that example, an antiderivative the x^3 is x^4/4, however x^4/4 + 2 is also one of one antiderivative. Despite, when we take it an unknown integral, we space in reality finding “all” the possible antiderivatives at when (as different values of C gives various antiderivatives). So there is ethereal difference in between them but they plainly are two various things. In additionally, we would say the a definite integral is a number which us could use the second component of the basic Theorem of Calculus; however an antiderivative is a duty which we could use the an initial part the the an essential 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”

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Sean Manoukian April 10, 2021 in ~ 11:04 am

Thanks for this, it’s really helpful. But I am wonder if over there is a typo in the last paragraph, here:

“For example, one antiderivative the x^3 is x^4/4”

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

Anyway, thanks!


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