HH_Sec_7_4 - 2 + 4 = 3 x + 1 x 2 + 4 i 3 x 3 + 12 x + 1 x 2...

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(9/30/08) Math 10B. Lecture Examples. Section 7.4. Algebraic identities and trigonometric substitutions Example 1 Find the partial-fraction decomposition of 1 x ( x - 1 ) . Check by giving the result a common denominator. Answer: 1 x ( x - 1) = 1 x - 1 - 1 x Check: 1 x - 1 - 1 x = x x ( x - 1) - x - 1 x ( x - 1) = x - ( x - 1) x ( x - 1) = 1 x ( x - 1) Example 2 Use the result of Example 1 to ±nd the function G ( x ) such that G p ( x ) = 1 x ( x - 1 ) and G ( 2 ) = 0 . Answer: G ( x ) = ln | x - 1 | - ln | x | + ln(2) Example 3 Find the partial-fraction decomposition of y = 3x 2 - 1 x 3 - x . Answer: 3 x 2 - 1 x 3 - x = 1 x + 1 x - 1 + 1 x + 1 Example 4 Use the result 3x 2 - 1 x 3 - x = 1 x + 1 x - 1 + 1 x + 1 of Example 3 to perform the integration, i 3x 2 - 1 x 3 - x dx . Answer: i 3 x 2 - 1 x 3 - x dx = ln | x | + ln | x - 1 | + ln | x + 1 | + C Example 5 Evaluate i 2 1 x 3 + 1 x 2 dx . Answer: i 2 1 x 3 + 1 x 2 dx = 2 Example 6 Find the antiderivative i 3x 3 + 12x + 1 x 2 + 4 dx . Answer: Figure A6 3 x 3 + 12 x + 1 x
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Unformatted text preview: 2 + 4 = 3 x + 1 x 2 + 4 i 3 x 3 + 12 x + 1 x 2 + 4 dx = 3 2 x 2 + 1 2 tan 1 ( 1 2 x ) + C 3 x x 2 + 4 3 x 3 + 12 x + 1 3 x 3 + 12 x 1 Figure A6 Interactive Examples Work the following Interactive Examples on Shenks web page, http//www.math.ucsd.edu/ashenk/: Section 8.4: Examples 1 through 4 Lecture notes to accompany Section 7.4 of Calculus by Hughes-Hallett et al. The chapter and section numbers on Shenks web site refer to his calculus manuscript and not to the chapters and sections of the textbook for the course. 1...
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