Application and Antiderivative
Application and Antiderivative
Antiderivates can be defined as the inverse function of derivatives. An antiderivative
of a function f(x) is a function whose derivative is f(x).
Some of the important formulas of Antiderivatives are as follows:-

(i) Let f (x) be function of x, then definite integral g of f (x) with respect to x between
the limit a & b is devoted by and defined by = which also known as limit of a sum.

(ii) The area bounded by the curve y = f (x), x-axis and x = a and x = b is given by
(iii) The area bounded by the curve x = f (y), y-axis and y = a & y = b is
(iv) Area between two curves and x = a & x = b is given by if the graphical about
both axes then,
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In calculus, an "anti-derivative", antiderivative, primitive integral or indefinite
integral[1] of a function f is a function F whose derivative is equal to f, i.e., F = f.[2][3]
The process of solving for antiderivatives is called antidifferentiation (or indefinite
integration) and its opposite operation is called differentiation, which is the process of
finding a derivative. Antiderivatives are related to definite integrals through the
fundamental theorem of calculus: the definite integral of a function over an interval is
equal to the difference between the values of an antiderivative evaluated at the
endpoints of the interval.
The function F(x) = x3/3 is an antiderivative of f(x) = x2. As the derivative of a
constant is zero, x2 will have an infinite number of antiderivatives; such as (x3/3) + 0,
(x3/3) + 7, (x3/3) - 42, (x3/3) + 293 etc. Thus, all the antiderivatives of x2 can be
obtained by changing the value of C in F(x) = (x3/3) + C; where C is an arbitrary
constant known as the constant of integration. Essentially, the graphs of
antiderivatives of a given function are vertical translations of each other; each graph's
location depending upon the value of C.
In physics, the integration of acceleration yields velocity plus a constant. The constant
is the initial velocity term that would be lost upon taking the derivative of velocity
because the derivative of a constant term is zero. This same pattern applies to further
integrations and derivatives of motion (position, velocity, acceleration, and so on).
Antiderivatives are also called general integrals, and sometimes integrals. The latter
term is generic, and refers not only to indefinite integrals (antiderivatives), but also to
definite integrals. When the word integral is used without additional specification, the
reader is supposed to deduce from the context whether it is referred to a definite or
indefinite integral.
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Some authors define the indefinite integral of a function as the set of its infinitely
many possible antiderivatives. Others define it as an arbitrarily selected element of
that set. Wikipedia adopts the latter approach.
Applications of Antiderivatives
In this section we will discuss two basic applications of antidi e
ff rentiation.
Antiderivatives and Di e
ff rential Equations
Antidifferentiation can be used in finding the general solution of the dif-
ferential equation.
Motion along a Straight Line
Antidifferentiation can be used to find specific antiderivatives using initial
conditions, including applications to motion along a line.


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