text: HH_Sec_7_1
  
  # HH_Sec_7_1 - (9/30/08) Math 10B. Lecture Examples. Section...
  
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  Unformatted text preview: (9/30/08) Math 10B. Lecture Examples. Section 7.1. Integration by substitution† Example 1 Find the antiderivative Answer: (x2 + 1)5 (2x) dx. 1 (x2 + 1)5 (2x) dx = 6 (x2 + 1)6 + C Example 2 Perform the integration Answer: x3 x4 + 16 dx. √ √ x3 x4 + 16 dx = 14 (x 6 + 16)3/2 + C Example 3 Find the antiderivative Answer: √ cos( x) √ dx. x √ √ cos( x) x +C dx = 2 sin √ x Example 4 Find the area of the region in Figure 1. y 3 2 1 y= 10x (x2 + 1)2 FIGURE 1 1 2 3 x Answer: [Area] = 9 2 1 Example 5 Evaluate 0 1 sin(π x) dx. 2 Answer: 0 sin(πx) dx = 1 π Example 6 Evaluate 0 1 e−2x dx. 1 (1 − e−2 ) 2 Answer: 0 e−2x dx = Interactive Examples Work the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/:‡ Section 6.8: Examples 1 through 5 Section 7.1: Example 4 notes to accompany Section 7.1 of Calculus by Hughes-Hallett et al. chapter and section numbers on Shenk’s web site refer to his calculus manuscript and not to the chapters and sections of the textbook for the course. ‡ The † Lecture 1 ...
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  ## This note was uploaded on 09/13/2010 for the course MATH math 10b taught by Professor Reed during the Summer '10 term at UCSD.
  
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explanation: The provided text is a set of lecture examples from a Math 10B course, specifically from Section 7.1, which focuses on integration by substitution. The document contains multiple math questions along with their corresponding answers.
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distilabel_metadata.content: HH_Sec_7_1
  
  # HH_Sec_7_1 - (9/30/08) Math 10B. Lecture Examples. Section...
  
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  Unformatted text preview: (9/30/08) Math 10B. Lecture Examples. Section 7.1. Integration by substitution† Example 1 Find the antiderivative Answer: (x2 + 1)5 (2x) dx. 1 (x2 + 1)5 (2x) dx = 6 (x2 + 1)6 + C Example 2 Perform the integration Answer: x3 x4 + 16 dx. √ √ x3 x4 + 16 dx = 14 (x 6 + 16)3/2 + C Example 3 Find the antiderivative Answer: √ cos( x) √ dx. x √ √ cos( x) x +C dx = 2 sin √ x Example 4 Find the area of the region in Figure 1. y 3 2 1 y= 10x (x2 + 1)2 FIGURE 1 1 2 3 x Answer: [Area] = 9 2 1 Example 5 Evaluate 0 1 sin(π x) dx. 2 Answer: 0 sin(πx) dx = 1 π Example 6 Evaluate 0 1 e−2x dx. 1 (1 − e−2 ) 2 Answer: 0 e−2x dx = Interactive Examples Work the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/:‡ Section 6.8: Examples 1 through 5 Section 7.1: Example 4 notes to accompany Section 7.1 of Calculus by Hughes-Hallett et al. chapter and section numbers on Shenk’s web site refer to his calculus manuscript and not to the chapters and sections of the textbook for the course. ‡ The † Lecture 1 ...
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  ## This note was uploaded on 09/13/2010 for the course MATH math 10b taught by Professor Reed during the Summer '10 term at UCSD.
  
  Ask a homework question - tutors are online
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The provided text is a set of lecture examples from a Math 10B course, specifically from Section 7.1, which focuses on integration by substitution. The document contains multiple math questions along with their corresponding answers.
  
  Label: multiple-question-and-answer