![]() ![]() Setting the condition f (2 a − x ) = f ( x) in equation (1), we getĠ ∫ 2 a f ( x) dx = a ∫ 0 dx = 2 a ∫ 0 f ( x) dx. When x = − a, we get u = a when x = 0, we get u = 0. In this integral, let us make the substitution, x = − u. ![]() (Recall that a function f ( x) is an odd function if and only if f ( − x ) = − f ( x). If f ( x) is an odd function, then − a ∫ a f ( x) dx = 0. − a ∫ a f ( x) dx = 0 ∫ a f ( x) dx + 0 ∫ a f ( x) dx = 2 0 ∫ a f ( x) dx . Substituting equation (2) in equation (1), we get When x = − a, we get u = a , when x = 0, we get u = 0, So, we get In the integral 0 ∫ − a f ( x) dx , let us make the substitution, x = − u. − a ∫ a f ( x) dx = − a ∫ 0 f ( x) dx + 0 ∫ a f ( x) dx . ![]() (Recall that a function f ( x) is an even function if and only if f ( − x ) = f ( x). If f ( x) is an even function, then − a ∫ a f ( x) dx = 2 0 ∫ a f ( x) dx. Replace a by 0 and b by a in the above property we get the following property ∴ b ∫ a f ( x) dx = a ∫ b f ( a + b − u )( − du ) = b ∫ a f ( a + b − u ) du We derive some more properties of definite integrals.ī ∫ a f ( x ) dx = b ∫ a f ( a + b − x ) dx Next, we give examples to illustrate the application of Property 5. īut, it is continuous in each of the sub-intervals that is, it is piece-wise continuous on. We note that the above function is not continuous on. In other words, it is defined by = n , if n ≤ x < ( n + 1), where n is an integer. We know that the greatest integer function is the largest integer less than or equal to x. We illustrate the use of the above properties by the following examples.Įvaluate : 1.5∫ 0 dx, where is the greatest integer function. This property is used for evaluating definite integrals by making substitution. I.e., the value of the definite integral changes by minus sign if the limits are interchanged. I.e., definite integral is independent of the change of variable. The value of a definite integral is unique.īy the second fundamental theorem of integral calculus, the following properties of definite integrals hold. As a short-hand form, we write F ( b) − F ( a) = b a. Since F ( b) − F ( a) is the value of the definite integral (Riemann integral) b ∫ a f ( x) dx, any arbitrary constant added to the anti-derivative F ( x) cancels out and hence it is not necessary to add an arbitrary constant to the anti-derivative, when we are evaluating definite integrals. If f ( x) be a continuous function defined on a closed interval and F ( x) is an anti derivative of f ( x), then, Theorem 9.2 (Second Fundamental Theorem of Integral Calculus) In other words, F ( x) is an anti-derivative of f ( x). If f ( x) be a continuous function defined on a closed interval and F ( x) = x ∫ a f ( u) du, a < x < b then, d/dx F ( x) = f ( x). Theorem 9.1 (First Fundamental Theorem of Integral Calculus) We state below the above important theorems without proofs. In fact, the two theorems provide a link between differential calculus and integral calculus. These theorems establish the connection between a function and its anti-derivative (if it exists). Their method is based upon two celebrated theorems known as First Fundamental Theorem and Second Fundamental Theorem of Integral Calculus. Both Newton and Leibnitz, more or less at the same time, devised an easy method for evaluating definite integrals. We observe in the above examples that evaluation of b ∫ a f ( x ) dx as a limit of the sum is quite tedious, even if f ( x) is a very simple function. Fundamental Theorems of Integral Calculus and their Applications ![]()
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