Indefinite Integral
An indefinite integral is a fundamental concept in calculus that represents the family of antiderivatives of a given function. It is denoted by the symbol ∫ and is used to find the general form of a function whose derivative is known. The result of an indefinite integral is expressed as a function plus an arbitrary constant.
7 Key excerpts on "Indefinite Integral"
eBook - ePub- Arsen Melkumian(Author)
- 2012(Publication Date)
- Routledge(Publisher)
...9 Indefinite and definite integrals Integrals play a twofold role in calculus. The so-called Indefinite Integral is an operation inverse to differentiation: the integral of function f (x) is a function whose derivative gives us f (x). We have seen the importance of derivatives throughout the textbook, and naturally the inverse operation also plays an important role in mathematical and economic analysis. Later, in Chapter 12, we study relations between quantities and their rates of change – the so-called differential equations. Solving differential equations is impossible without Indefinite Integrals. The other type of integral is the definite integral. Geometrically a definite integral represents the area under a curve. But in applications the meaning of integration is totaling continuous quantities. For instance, to reconstruct profit from its rate of change we would integrate the latter function. Another application of integrals is in studying consumer–producer surplus. 9.1 Indefinite Integrals If f (x) and F(x) are some functions of x such that then F(x) is called an antiderivative of f (x). For example, the function F 1 (x) = 3 x 2 + 5 is an antiderivative of f (x) = 6 x. Note that F 1 (x) = 3 x 2 + 5 is not the only antiderivative of f (x). In fact, the function f (x) = 6 x has infinitely many antiderivatives and all antiderivatives of f (x) are functions of the form where C is a constant. We will refer to F(x) = 3 x 2 + C as the general antiderivative of f (x) and we will write to indicate that. The symbol is referred to as the integral sign, 6 x is the integrand and C is the constant of integration. The expression is known as the Indefinite Integral of f (x) = 6 x. T HEOREM 9.1: Let f (x) be a differentiable function of x, and k, n and C be some constants...
eBook - ePub- Richard C. Dorf, Ronald J. Tallarida(Authors)
- 2018(Publication Date)
- CRC Press(Publisher)
...7 Integral Calculus 1. Indefinite Integral If F (x) is differentiable for all values of x in the interval (a, b) and satisfies the equation dy / dx = f (x), then F (x) is an integral of f (x) with respect to x. The notation is F (x) = ∫ f (x) dx or, in differential form, F (x) = f (x) dx. For any function F (x) that is an integral of f (x) it follows that F (x) + C is also an integral. We thus write ∫ f (x) d x = F (x) + C. (See Table of Integrals.) 2. Definite Integral Let f (x) be defined on the interval [ a, b ] which is partitioned by points x 1, x 2, …, x j, …, x n − 1 between a = x 0 and b = x n. The j th interval has length Δ x j = x j − x j −1, which may vary with j. The sum, ∑ j = 1 n f (v j) Δ x j,, where υ j is arbitrarily chosen in the j th subinterval, depends on the numbers x 0,…, x n and the choice of the υ as well as f ; but if such sums approach a common value as all Δ x approach zero, then this value is the definite integral of f over the interval a, b) and is denoted ∫ a b f (x) d x. The fundamental theorem of integral calculus states that ∫ a b f (x) d x = F (b) − F (a), where F is any continuous Indefinite Integral of f in the interval (a,. b). 3. Properties ∫ a b [ f 1 (x) + f (x) + ⋯ + f j (x) ] d x = ∫ a b f 1 (x) d x + ∫ a b f 2 (x) d x + ⋯ + ∫ a b f j (x) d x. ∫ a b c f (x) d x = c ∫ a b f (x.) d x. if c is a constant. ∫ a b f (x) d x = − ∫ b a f (x) d x. ∫ a b f (x) d x = ∫ a c f (x) d x + ∫ c b f (x) d x. 4. Common Applications of the Definite Integral • Area (Rectangular Coordinates) Given the function y = f (x) such that y > 0 for. all x between a and b, the area bounded by the curve y = f (x), the x-axis, and the vertical lines x = a and x = b is A = ∫ a b f (x) d x. • Length of Arc (Rectangular Coordinates) Given the smooth curve f (x, y) = 0 from point (x 1, y 1) to point (x 2, y 2), the length between these points...
- F. Xavier Malcata(Author)
- 2020(Publication Date)
- Wiley(Publisher)
...(11.1), the (indefinite) integral is the inverse function of the derivative, i.e. (11.5) which may be restated as (11.6) with the aid of Eq. (11.1) and hypothesis ; either Eq. (11.5) or Eq. (11.6) justifies why composition of integration and differentiation of a given function retrieves the original function. Unlike the derivative of an elementary function – which normally exists and is another elementary function (as long as the function under scrutiny is continuous), the same cannot be stated about an integral; there are elementary functions that do not hold an integral expressible as a finite combination of elementary functions (e.g.) – and furthermore it cannot be told a priori whether a given function can be integrated. One also realizes that the uniqueness of the derivative does not extend to an integral. Consider, in this regard, that F 1 { x } and F 2 { x } represent two distinct (indefinite) integrals of f { x } – both continuous and differentiable functions; according to Eq. (11.6), this means that (11.7) as well as (11.8) Ordered subtraction of Eq. (11.7) from Eq. (11.8) produces (11.9) where the rule of differentiation of an (algebraic) sum as per Eq. (10.106) allows reformulation to (11.10) Lagrange’s theorem may be retrieved here because its conditions of validity are met – namely, Φ { x } ≡ F 2 { x } − F 1 { x } is a continuous function within interval [ a,b ], since both F 1 { x } and F 2 { x } are so separately, within the same interval, as per Eq (9.139) ; according to Eq. (10.274), for an arbitrary x comprised between a and b (and thus defining an interval] a,x [), one can always find a ξ ∈] a,x [ such that (11.11) However, Eq. (11.10) enforces a nil value for dΦ { x }/ dx, irrespective of the actual value taken by x – so Eq. (11.11) becomes (11.12) Eq...
eBook - ePub- Mike Aitken, Bill Broadhurst, Stephen Hladky(Authors)
- 2009(Publication Date)
- Garland Science(Publisher)
...Just as we saw with Leibniz’s d y /d x notation for the derivative in Section 5.3, the ‘d x ’ symbol here is the mathematical abbreviation for ‘an infinitesimal increment in x ’. In words, an equation of the sort shown in Figure 6.9 tells us that ‘the integral of 2 x with respect to x is x 2 + c ’. Note that the function being integrated, 2 x in this case, is called the integrand. Technically, this whole expression is termed an Indefinite Integral to help distinguish it from the definite integrals that we will encounter shortly in Section 6.3. Figure 6.9 An Indefinite Integral consists of an integral sign, an integrand, and a differential. Another way of describing the Indefinite Integral of a function f (x) is as an ‘antiderivative function’, which is often written using the same label, but in upper case: F (x). This allows us to write the defining feature of the antiderivative concisely as: d F (x) d x = f (x). (EQ6.1) This means that an antiderivative is any function that when differentiated takes us back to the original function f (x). The number of antiderivative functions is infinite, but they are all very closely related. If we know one of them, we can obtain any of the others simply by adding a constant to it. The new function will also be an antiderivative of f (x) because the derivative of any constant term disappears. So, if F (x) is one of the antiderivatives of the function f (x), we can use it to write a general expression for an Indefinite Integral: ∫ f (x) d x = F (x) + c. (EQ6.2) We will apply these ideas straightaway by attempting to reconstruct the antiderivative of the function r = 0.006 t 0.5, in which r (measured in milligrams per hour) represents the rate at which the mass of a kidney stone increases with time t (in hours). The integral of this expression with respect to time will make it possible to calculate the total mass of the crystalline aggregate...
- J. Rosebush, Stu Schwartz(Authors)
- 2016(Publication Date)
- Research & Education Association(Publisher)
...PART IV INTEGRALS Chapter 12 Types of Integrals, Interpretations and Properties of Definite Integrals, Theorems I. TYPES OF INTEGRALS A. Indefinite Integrals have no limits, ∫ f (x) dx. This represents the antiderivative of f (x). That is, if ∫ f (x) dx = F (x) + C, then F ′(x) = f (x). When taking an antiderivative of a function, don’t forget to add C ! For instance, ∫2 xdx = x 2 + C (The constant C is necessary because the antiderivative of f (x) = 2 x could be F (x) = x 2 or F (x) = x 2 + 1 or F (x) = x 2 – 2, and so on.) Sometimes, you are given an initial condition that allows you to find the value of C. For instance, find the antiderivative, F (x), of f (x) = 2 x, given that F (0) = 1. Then, F (x) = ∫2 xdx = x 2 + C → F (0) = (0) 2 + C = 1 → C = 1 → F (x) = x 2 + 1. Another way of posing this question is: Find y if and y | x =0 = 1. The equation is called a differential equation (more on this later) because it contains a derivative. B. Definite integrals have limits x = a and. If f (x) is continuous on [ a, b ] and F ′(x) = f (x), then (The First Fundamental Theorem of Calculus.) 1. A definite integral value could be positive, negative, zero or infinity. When used to find area, the definite integral must have a positive value. i. If f (x) > 0 on [ a, b ], then and geometrically it represents the area between the graph of f (x) and the x -axis on the interval [ a, b ]. For example, square units. Note that this could also have been solved geometrically because the area in question is that of a right triangle with a base of 3 units and a height of 6 units. Solving an area problem geometrically is really helpful when the question involves the integral of a piecewise linear function, for instance,. This represents the area between the function f (x) = | x | and the x -axis between x = –1 and x = 3...
eBook - ePub- Gregory Hill, Mel Friedman(Authors)
- 2013(Publication Date)
- Research & Education Association(Publisher)
...The result does not depend on whether a and b are positive or negative. The left base is 2 a – 1 and the right base is 2 b – 1. The height is b – a. The sum of the two results is b 3 – a 3 + b 2 – a 2 – b + a. 6.6 FUNDAMENTAL THEOREM OF CALCULUS One of the most impressive developments in calculus was the connection between the differential and integral branches found in what came to be called the Fundamental Theorem of Calculus. One part of the Fundamental Theorem asserts that a definite integral of a continuous function is a function of its upper limit and is therefore itself differentiable. A second part of the Fundamental Theorem establishes how to analytically evaluate a definite integral. Both parts will be presented here without formal proof, but if it enhances your understanding, any online search for the proof of the Fundamental Theorem of Calculus will produce an abundance of sources to read and study. Fundamental Theorem of Calculus (Part 1) If f is a continuous function, a is a constant, and then This theorem essentially says, “The instantaneous rate of change of accumulation, or loss, of area between a function and the x- axis at any point, is equal to the function value of the integrand at that point.” Let’s take an intuitive way to explain what is happening, using a discrete approach to the concept of accumulating area under a curve. Think of a definite integral as a Riemann sum over a given interval. As the upper limit of the given interval changes, accumulated area increases or decreases by “adding” another infinitely thin rectangle to the previously summed rectangles. If the function values are “large” at a particular x value, then the next rectangle added will add more area than if the function values are small. It follows logically that the amount of change in accumulated area is related to the magnitude of the function being integrated. Notice that the upper limit is simply x...
- Magno Urbano(Author)
- 2019(Publication Date)
- Wiley(Publisher)
...2 Infinitesimal Calculus : A Brief Introduction 2.1 Introduction In this chapter we will do a brief introduction to infinitesimal calculus or differential and integral, known as simply, calculus. Wikipedia has as good definition about calculus: Calculus is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. It has two major branches, differential calculus (concerning instantaneous rates of change and slopes of curves) and integral calculus (concerning accumulation of quantities and the areas under and between curves). These two branches are related to each other by the fundamental theorem of Calculus. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well‐defined limit. Generally, modern calculus is considered to have been developed, independently, in the 17 th century by Isaac Newton and Gottfried Wilhelm Leibniz. Today, Calculus has widespread uses in science, engineering, and economics. Calculus is a part of modern mathematics education. 2.2 The Concept Behind Calculus Suppose we have a car traveling in a straight line for 1 h, at a constant speed of 100 km/h, and later reducing the speed in half and traveling for another hour. What is the average speed of that car? The answer is Now let us see a more complex problem. The car travels 15 min at a nonconstant speed of 15 km/h, stops for 5 min, travels for 5 km at a nonconstant speed of 8 km/h, stops again, and then travels a distance of 2 km at 15 km/h. What is the average speed now? The answer is not evident because the method we have cannot deal with variable entities...
