Chapter6(1804) - Ch6/MATH1804/YMC/2009-10 1 Chapter 6...

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Ch6/MATH1804/YMC/2009-10 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 0 ( 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 .
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Ch6/MATH1804/YMC/2009-10 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 x dx = - cos x + C Z cos x dx = sin x + C Z sec 2 x dx = tan x + C Z sec x tan x dx = sec x + C Z csc 2 x dx = - cot x + C Z csc x cot x dx = - 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 x -axis, and the lines x = a and x = b
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Ch6/MATH1804/YMC/2009-10 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. We now state the Fundamental Theorem of Calculus in two parts: Theorem 6.4 ( Fundamental Theorem of Calculus (I) ) Let f : [ a, b ] -→ R be continuous on [ a, b ] .
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