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The derivative can be defined as the slope of a tangent line. When taking a derivative the general formula to follow would be: Constant Rule $\frac{d(c)}{dx}=0$ The antiderivative, also referred to as an integral, can be thought of as the inverse operation for the derivative. In other words, it is the opposite of a derivative.
i think that indefinite integral and anti derivative are very much closely related things but definitely equal to each other. indefinite integral denoted by the symbol"∫" is the family of all the anti derivatives of the integrand f(x) and anti derivative is the many possible answers which may be evaluated from the indefinite integral. e.g ...
The basis for the names "primitive" and "derivative" is just their roles in deriving the derivative function from the primitive function. There seem to be no books on books.google.com before 1900 that contain the term "antiderivative." The earliest reference there is, oddly enough, a course description in the catalog of Univ. Chicago for 1903/04.
$\begingroup$ I upvoted this question because in my opinion, it's a real question because some mathematicians have a demand for rigour ad just like the Jordan curve theorem, it seems so obviously true that an integral is an antiderivative but is probably actually quite difficult to give a rigorous proof of.
$\begingroup$ I think the point of confusion is that the tool itself uses two different tools to compute the anti-derivative. One uses a tool called "Maxima" to do so. The other uses its own methodology which also provides steps.
As we know, a non-continuous function may have an antiderivative. Thus, the function may not be integrable. That is, although there is an explicitly defined antiderivative F(x) (possibly not, but just expressed as an integral using the Fundamental Theorem of Calculus), the limit of the Riemann sums may not exit or may not equal to some finite ...
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.
antiderivative --- one function. indefinite integral --- set of functions. definite integral --- number. As mentioned in a comment, antiderivatives and definite integrals are related by (a version of) the Fundamental Theorem of Calculus.
Think of a simpler example: if all we have available as “elementary functions” are polynomials or, more generally, rational functions, the function $1/x$ wouldn't admit an “elementary antiderivative”, but it still would have one: $$ \int_{1}^{x}\frac{1}{t}\,dt $$ Since this is a “new” function, we give it a name, precisely “ $\log ...
So if you're gonna declare variables for a first antiderivative, you might as well do it for antiderivatives of all orders. $\endgroup$ – Git Gud Commented Apr 25, 2014 at 20:17