Most of civilization have a misunderstanding of the relationship between “integration” and “taking antiderivative”; they tend 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 original function, i m sorry is characterized by a limiting process. Let’s small “integration” down an ext precisely right into two parts, 1) unknown integral and also 2) definite integral. Unknown integral means integrating a function without any limit yet in definite integral there room upper and lower limits, in the various other words we called that the expression of integration.
While one antiderivative just method that to uncover the functions whom derivative will certainly be our initial function. Over there is a very small difference in between definite integral and antiderivative, however there is clearly a big difference in between indefinite integral and also antiderivative. Let’s consider an example:
f(x) = x²
The antiderivative the x² is F(x) = ⅓ x³.
The indefinite integral is ∫ x² dx = F(x) = ⅓ x³ + C, i beg your pardon is nearly the antiderivative other than c. (where “C” is a continuous number.)
On the other hand, us learned around the fundamental Theorem that Calculus couple weeks ago, whereby we need to apply the second component of this theorem in to a “definite integral”.
The definite integral, however, is ∫ x² dx native a come b = F(b) – F(a) = ⅓ (b³ – a³).
The indefinite integral is ⅓ x³ + C, since the C is undetermined, for this reason this is not just a function, rather it is a “family” that functions. Deeply reasoning an antiderivative of f(x) is simply any role whose derivative is f(x). For example, one antiderivative the x^3 is x^4/4, but x^4/4 + 2 is likewise one of one antiderivative. Despite, as soon as we take an indefinite integral, we space in reality finding “all” the feasible antiderivatives at when (as different values that C gives various antiderivatives). So over there is subtle difference in between them however they clearly are two various things. In additionally, we would say that a identify integral is a number which us could apply the second part of the basic Theorem that Calculus; however an antiderivative is a role which us could use the first part of the basic Theorem of Calculus.
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This entry was posted in Uncategorized top top January 25, 2017 by moiz ali.
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5 think on “Integral vs Antiderivative”
Thanks for this, it’s very helpful. Yet I am wondering if there is a typo in the final paragraph, here:
“For example, one antiderivative that x^3 is x^4/4”
Shouldn’t the be 1/4 x^4 rather of x^4/4?
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