SalasSV_08_01 - CHAPTER 4 THE MEAN-VALUE THEOREM AND...

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Unformatted text preview: CHAPTER 4 THE MEAN-VALUE THEOREM AND APPLICATIONS CHAPTER 8 TECHNIQUES OF INTEGRATION 8.1 INTEGRAL TABLES AND REVIEW We begin by listing the more important integrals with which you are already familiar. 1. k dx = kx + C , k constant. 2. x r dx = 1 r + 1 x r + 1 + C , r = 1. 3. dx x = ln | x | + C . 4. e x dx = e x + C . 5. p x dx = p x ln p + C , p > 0 constant, p = 1. 6. sin x dx = cos x + C . 7. cos x dx = sin x + C . 8. tan x dx = ln | sec x | + C . 9. cot x dx = ln | sin x | + C . 10. sec x dx = ln | sec x + tan x | + C . 11. csc x dx = ln | csc x cot x | + C . 12. sec x tan x dx = sec x + C . 13. csc x cot x dx = csc x + C . 14. sec 2 x dx = tan x + C . 15. csc 2 x dx = cot x + C . 16. dx a 2 x 2 = sin 1 x a + C , a > 0 constant. 17. dx a 2 + x 2 = 1 a tan 1 x a + C , a > 0 constant. 446 8.1 INTEGRAL TABLES AND REVIEW 447 18. dx x x 2 a 2 = 1 a sec 1 | x | a + C , a > 0 constant. 19. sinh x dx = cosh x + C . 20. cosh x dx = sinh x + C . For review we work out a few integrals involving u-substitutions. Example 1 Find x tan x 2 dx . SOLUTION Set u = x 2 , du = 2 x dx . Then x tan x 2 dx = 1 2 tan u du = 1 2 ln | sec u | + C = 1 2 | sec x 2 | + C . Formula 8 Example 2 Calculate 1 e x e x + 2 dx . SOLUTION Set u = e x + 2, du = e x dx . At x = 0, u = 3; at x = 1, u = e + 2. Thus 1 e x e x + 2 dx = e + 2 3 du u = ln | u | e + 2 3 Formula 3 = ln ( e + 2) ln 3 = ln 1 3 ( e + 2) = 0. 45. Example 3 Find cos 2 x (2 + sin 2 x ) 1 / 3 dx ....
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This note was uploaded on 10/12/2010 for the course MATH 12345 taught by Professor Smith during the Spring '10 term at University of Houston - Downtown.

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SalasSV_08_01 - CHAPTER 4 THE MEAN-VALUE THEOREM AND...

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