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.
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.
this as very helpful thanks a lot
Thanks for the post, that clarifies a bunch (-:
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!
1/4 x^4 =(1/4) * (x^4)/(1) = (1)(x^4) / (4)(1) = (x^4) / (4) = x^4/4.
no the formula is x^n= x^n+1/n+1
if we take 1/(4x^4) and we Derivative we get -1/x^5+c
,x^4/4 is x^3+c
Awesome, thanks for the post.
Thank you for the article.
re: “there is clearly a big difference in between indefinite integral and antiderivative”
To me, it looks like that there is more of a difference between the DEFINITE integration and the antiderivative. In your example of x²:
(a) the indefinite integral is ⅓x³ + C.
(b) the definite integral from a to b = F(b) – F(a) = ⅓ (b³ – a³).
(c) the antiderivative is from ⅓x³
Isn’t (a) more similar to (c) than it is to (b)?
You may be a math wizard, but your English needs work. Typical.
Agreed. Someone should have proof read this. It’s distracting.
Most of the time, the derivative looks like they are reversible, including the constants
dx^2 = 2x^1 ∫ 2x ^1 = x^2 +C
dx^1 = 1X^0 ∫ x^0 = x + C
dx^0 = 0 ∫ x^-1 = ln(x) + C <—- This one is not like the others
dx^-1 = -1X^-2 ∫ -x^-2 = x^-1 + C
I think you are incorrect in stating that the antiderivative of x^2 is 1/3x^1/3. It should be F(x) = 1/3x^1/3 + C, as it is the most general antiderivative which would apply to any x within the function’s bounds. Unless I am mistaken, the indefinite integral is the same as the general antiderivative, contrary to what you state here.
I) d/dx of x^n= x^(n-1)
Thus,
II) f(x)dx for x^n=[x^(n+1)]/n+1
Test that with x^3 for the derivative first then the anti derivative of that result. i.e d/dx = 3x^2 but applying rule 2 to this derivative takes us back to x^3 that’s [3x^(2+1)]/2+1= x^3. I hope you get now