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Unformatted text preview: Ch6/MATH1804/YMC/200910 1 Chapter 6. Integration 6.1. The Fundamental Theorem of Calculus The principal theorem of this section is the Fundamental Theorem of Calculus , which is the central theorem of integral calculus. It provides a connection between the operations of differentiation and integration, enabling us to compute integrals using an antiderivative of the integrand function. Definition 6.1 An antiderivative of the function f is a function F such that F ( x ) = f ( x ) Example 6.2 function f ( x ) 1 2 x cos x sin 2 x + 4 x 3 sec 2 x antiderivative F ( x ) x x 2 sin x 1 2 cos 2 x + x 4 tan x Remark 6.3 Note that if F 1 and F 2 are antiderivatives of f , then F 1 ( x ) F 2 ( x ) = C , where C ∈ R . In other words, if F is an antiderivative of f , then every antiderivative of f has the form F ( x ) + C where C is an arbitrary constant. We call the function F ( x ) + C the indefinite integral of f ( x ) , and we write Z f ( x ) dx = F ( x ) + C where C is a constant Properties of the Indefinite Integration 1. Z k dx = kx + C where k is constant. 2. Z kf ( x ) dx = k Z f ( x ) dx where k is constant. 3. Z ( f ( x ) ± g ( x )) dx = Z f ( x ) dx ± Z g ( x )) dx . Ch6/MATH1804/YMC/200910 2 Basic Integration Formulas 1. Z x n dx = x n +1 n + 1 + C for n 6 = 1 . 2. Z 1 x dx = Z dx x = ln  x  + C . 3. Z e x dx = e x + C Integration Formulas for Trigonometric Functions Z sin xdx = cos x + C Z cos xdx = sin x + C Z sec 2 xdx = tan x + C Z sec x tan xdx = sec x + C Z csc 2 xdx = cot x + C Z csc x cot xdx = csc x + C The definite integral of f ( x ) over the interval [ a,b ] is denoted by Z b a f ( x ) dx. The numbers a and b are called limits of integration ; a is the lower limit and b is the upper limit . Suppose that the graph of f ( x ) over the interval [ a,b ] looks like the following: The geometric interpretation of the definite integral is that Z b a f ( x ) dx = Area of the region bounded by the curve y = f ( x ) , the xaxis, and the lines x = a and x = b Ch6/MATH1804/YMC/200910 3 Properties of Definite Integrals 1. Z b a f ( x ) dx = Z a b f ( x ) dx 2. Z a a f ( x ) dx = 0 3. Z b a k f ( x ) dx = k Z b a f ( x ) dx where k is any number 4. Z b a ( f ( x ) ± g ( x )) dx = Z b a f ( x ) dx ± Z b a g ( x ) dx 5. Z c a f ( x ) dx = Z b a f ( x ) dx + Z c b f ( x ) dx . This property means that the definite integral over an interval can be expressed in terms of definite integrals over subintervals....
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This note was uploaded on 11/07/2010 for the course SCIENCE math1804 taught by Professor Prof during the Spring '10 term at HKU.
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