MTH_112_Ch10_Antideri_revised_slides_Nov. 10_2011

MTH_112_Ch10_Antideri_revised_slides_Nov. 10_2011 -...

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Antiderivatives Semester 1 () Antiderivatives Semester 1 1 / 56

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Outline 1 Antiderivatives 2 Rules for Integration 3 Integration by substitution 4 Integration by parts 5 Some trigonometric integrals () Antiderivatives Semester 1 2 / 56
Outline 1 Antiderivatives 2 Rules for Integration 3 Integration by substitution 4 Integration by parts 5 Some trigonometric integrals () Antiderivatives Semester 1 3 / 56

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Antiderivatives DIFFERENTIATION AND INTEGRATION ARE INVERSE PROCESSES. We now consider the ‘inverse problem’ to differentiation: Given a function f , is there a function F such that F 0 ( x ) = f ( x ) ? If such a function F exists, it is called an antiderivative of f . The process of ﬁnding F ( x ) is called integration . () Antiderivatives Semester 1 4 / 56
Deﬁnition Deﬁnition A function F is said to be an antiderivative of f on an interval ( a , b ) if F 0 ( x ) = f ( x ) for all x in ( a , b ) . () Antiderivatives Semester 1 5 / 56

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Antiderivatives of f ( x ) = 1. Example Suppose f ( x ) = 1 on R . Then all the following functions are antiderivatives of f . F ( x ) = x + 1 , F ( x ) = x + 2 , F ( x ) = x + 3 , F ( x ) = x - 1 , F ( x ) = x - 1 2 , F ( x ) = x - π ··· . In fact, the function F ( x ) = x + C for some constant C is an antiderivative of f ( x ) = 1. () Antiderivatives Semester 1 6 / 56
General antiderivative Theorem If F is an antiderivative of f on an interval I,then the most general antiderivative of f on I is F ( x ) + C where C is any arbitrary constant. () Antiderivatives Semester 1 7 / 56

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Indeﬁnite Integral R f ( x ) dx We denote by R f ( x ) dx the most general derivative of f , and call it the indeﬁnite integral of f . So, if f ( x ) = x n ( n 6 = - 1 ) , then we have Z f ( x ) dx = Z x n dx = x n + 1 n + 1 + C . () Antiderivatives Semester 1 8 / 56
Integration By integration, we mean the process of ﬁnding F ( x ) , given that F 0 ( x ) = f ( x ) . We shall determine the indeﬁnite integral R f ( x ) dx . The function f is called the integrand . () Antiderivatives Semester 1 9 / 56

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Example Example For the function f ( x ) = 1 x , x 6 = 0, prove that Z f ( x ) dx = ln | x | + C . () Antiderivatives Semester 1 10 / 56
Solution First we note that it follows from the deﬁnition that Z f ( x ) dx = ln | x | + C ⇐⇒ d dx ln | x | = f ( x ) = 1 x . It sufﬁces for us to prove the latter statement. ln | x | = ( ln x if x > 0 , ln ( - x ) if x < 0 . () Antiderivatives Semester 1 11 / 56

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For x > 0, we have d dx ln | x | = d dx ln x = 1 x ; whereas for x < 0, we have d dx ln | x | = d dx ln ( - x ) = 1 - x · ( - 1 ) = 1 / x . Therefore, we have proven that
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MTH_112_Ch10_Antideri_revised_slides_Nov. 10_2011 -...

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