integrals notes

integrals notes - MATH 152 Applied Linear Algebra and...

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Unformatted text preview: MATH 152 Applied Linear Algebra and Differential Equations Fall 2006-07 2 Supplementary Notes – Indefinite Integrals: 3 Techniques September 1, 2006 1 Indefinite Integrals Definition Let f ( x ) be defined in [ a,b ] . A function F ( x ) defined in [ a,b ] is called a primitive function of f ( x ) in the interval if F ( x ) = f ( x ) for all x in [ a,b ] . In fact, when the relation d dx F ( x ) = f ( x ) is satisfied, we call the primitive function (or the antiderivative ) F ( x ) an indefinite integral of f ( x ) and denote F ( x ) = Z f ( x ) dx. We refer to f ( x ) as the integrand and to R as the integral sign . The symbol dx (known as differential ) is a label, indicating the variable with respect to which we are integrating. So we mean the integration is taken with respect to x . If F ( x ) is a primitive function of f ( x ), then F ( x )+ C ( C can be any constant) is a whole class of functions having the same derivative (why?). Therefore, using the integral sign defined above, we may denote Z f ( x ) dx = F ( x ) + C . The arbitrary constant C is called the constant of integration . ¥ Example 1 (Primitive function) A primitive function of f ( x ) = x 2 is F ( x ) = 1 3 x 3 since F ( x ) = f ( x ) = x 2 . Note that if F is a primitive function of f , so is F + C for any constant C since d dx ( F + C ) = F + 0 = f . Hence, primitive functions of x 2 can be 1 3 x 3 , 1 3 x 3 + 100, 1 3 x 3 + 1 2 , etc.. In general, the primitive function is not unique. Any two primitive functions differ with each other by a constant. Here we should write R x 2 dx = 1 3 x 3 + C . 2 ¥ Example 2 (Indefinite integration) 1. Z k dx = kx + C , k constant. ( d dx ( kx + C ) = k + 0 = k. ) 2. Z x n dx = x n +1 n + 1 + C , n 6 =- 1. ‡ d dx ‡ x n +1 n +1 + C · = x n + 0 = x n . · 3. Z 1 x dx = ln | x | + C , x 6 = 0. ˆ 1. x > , d dx (ln | x | + C ) = d dx ln x = 1 x . 2. x < , d dx (ln | x | + C ) = d dx ln(- x ) = 1- x d (- x ) dx = 1 x . ! 1 4. Z e x dx = e x + C . ( d dx ( e x + C ) = e x + 0 = e x . ) 5. Z kf ( x ) dx = k Z f ( x ) dx , k constant.    Let R f ( x ) dx = F ( x ) + C . Then F = f . Now d dx ( kF ( x ) + kC ) = kF ( x ) + 0 = kf ( x ). Then R kf ( x ) dx = kF ( x ) + kC = k R f ( x ) dx .    6. Z ( f ( x ) + g ( x )) dx = Z f ( x ) dx + Z g ( x ) dx.         Let R f ( x ) dx = F ( x ) + C 1 , R g ( x ) dx = G ( x ) + C 2 . Then R f ( x ) dx + R g ( x ) dx = F ( x ) + G ( x ) + C 1 + C 2 . Now d dx ( F ( x ) + G ( x ) + C 1 + C 2 ) = F ( x ) + G ( x ) + 0 = f ( x ) + g ( x ). Then R ( f ( x ) + g ( x )) dx = F ( x ) + G ( x ) + C 1 + C 2 = R f ( x ) dx + R g ( x ) dx .         7. Z x 2 x + 2 3 ¶ dx = Z x 3 + 2 3 x 2 ¶ dx = Z x 3 dx + 2 3 Z x 2 dx = x 4 4 + 2 9 x 3 + C....
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This note was uploaded on 09/30/2010 for the course MATH MATH152 taught by Professor Kcc during the Spring '10 term at HKUST.

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integrals notes - MATH 152 Applied Linear Algebra and...

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