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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, Know More About: Anti derivative chain rule Tutorcircle.com Page No. : 1/4 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. Read More About: Antiderivative of 0 Tutorcircle.com Page No. : 2/4 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 antidiﬀerentiation. Antiderivatives and Diﬀerential 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. Tutorcircle.com Page No. : 3/4 Page No. : 2/3 Thank You TutorCircle.com