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Unformatted text preview: (9/30/08) Math 10B. Lecture Examples. Section 7.2. Integration by parts†
Example 1 (a) Find the antiderivative diﬀerentiation.
Answer: (a) x cos x dx. (b) Check the result by
d (x sin x + cos x) = x cos x. dx x cos x dx = x sin x + cos x + C (b) Product Rule: Example 2 What is the area of the region between y = xe−x and the x-axis for 0 ≤ x ≤ 4? Answer: Figure A2 • [Area] = 1 − 5e−4 y 0.4 0.2 y = xe−x 2 Figure A2 4 x Example 3 Find the antiderivative
Answer: (a) x3 ln x dx. x3 ln x dx =
5 14 x 4 1 ln x − 16 x4 + C Example 4 Evaluate
5 ln (2x) dx. Answer:
1 ln(2x) dx = 5 ln(10) − ln(2) − 4 Example 5 Find the indeﬁnite integral
Answer: (ln x)4 x dx. (ln x)4 x dx = 1 (ln x)5 5 +C Interactive Examples Work the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/:‡ Section 8.1: Examples 1 and 2 notes to accompany Section 7.2 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.
- Summer '10