Most of world have a ofradiation of the relationship between “integration” and also “taking antiderivative”; they have tendency to say this words as synonyms, yet there is a slight difference.
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In general, “Integral” is a function associate through the initial function, i m sorry is defined by a limiting process. Let’s small “integration” down much more precisely right into two parts, 1) unknown integral and also 2) definite integral. Unknown integral means integrating a role without any kind of limit but in identify integral there are upper and lower limits, in the various other words we called that the interval of integration.
While an antiderivative just means that to uncover the functions whom derivative will be our initial function. There is a very small difference in in between definite integral and also antiderivative, but there is clearly a huge difference in 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 practically the antiderivative except c. (where “C” is a constant number.)
On the various other hand, us learned about the an essential Theorem the Calculus pair weeks ago, whereby we require to apply the second component of this to organize in come a “definite integral”.
The identify integral, however, is ∫ x² dx native a come b = F(b) – F(a) = ⅓ (b³ – a³).
The indefinite integral is ⅓ x³ + C, due to the fact that the C is undetermined, for this reason this is not only a function, instead it is a “family” of functions. Deeply reasoning an antiderivative of f(x) is simply any duty whose derivative is f(x). For example, an antiderivative the x^3 is x^4/4, but x^4/4 + 2 is additionally one of an antiderivative. Despite, as soon as we take it an unknown integral, we space in fact finding “all” the possible antiderivatives at when (as various values that C gives different antiderivatives). So there is subtle difference in between them however they clearly 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 an essential Theorem the Calculus; yet an antiderivative is a duty which we could apply the first part the the an essential Theorem of Calculus.
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5 think on “Integral vs Antiderivative”
Thanks because that this, it’s very helpful. But I am wondering if over there is a typo in the last paragraph, here:
“For example, one antiderivative that x^3 is x^4/4”
Shouldn’t that be 1/4 x^4 rather of x^4/4?
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