c2IntegrationTechniquea14 - Page 1 of 125 MA1201 Calculus...

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Essential Calculus
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Chapter 4 / Exercise 18
Essential Calculus
Stewart
Expert Verified
Page 1of 125MA1201 Calculus and Basic Linear Algebra II Chapter 2 Integration Technique
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Essential Calculus
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Chapter 4 / Exercise 18
Essential Calculus
Stewart
Expert Verified
Page 2of 125In this Chapter, we shall discuss some important technique in evaluating integrals: Method of Substitution (pp. 3 - 37) Integration by parts & Reduction Formula (pp. 38 - 89) Integration of rational functions Method of partial fractions (pp.90 - 110)
Page 3of 125Method of substitution The principle of this method is to “transform” some complicated integrals into some simpler integrals by introducing a new variable ?. (Difficult to integrate) (Easy to integrate) Procedure of transformation Let ? = ?3+ 1, then ????= 3?2⇒ ?? =13?2??.3?2?3+ 1??1??? = ln |?| + ?? = ?3+ 13?2?3+ 1??3?2?3+ 1(13?2??)= ∫1?3+ 1??1???Change ??to ??Express the integrand in terms of ?
Page 4of 125Example 1 Compute the integral ∫ ?2cos(?3+ 1) ??
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Page 6of 125Example 2 Compute the integral ∫ ?5?2+ 3??
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Page 8of 125Remark of Example 2 (On the choice of substitution)One may try to transform the integral by using the substitution ? = √?2+ 3. The computation is more tedious although one can obtain the answer: Therefore, one should choose some simple (but effective) substitution in order to minimize the computation cost. Remember, “simple is the best”.Note that ????=12√?2+3(2?) =?√?2+3, then the integral becomes ∫ ?5?2+ 3?? = ∫ ?5?2+ 3√?2+ 3??? = ∫ ?4(?2+ 3)??= ∫(?2− 3)2?2?? = ∫(?6− 6?4+ 9?2)?? =?776?55+ 3?3+ ?=17(?2+ 3)7265

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