thomasET_226348_ism39

thomasET_226348_ism39 - CHAPTER 7 INTEGRALS AND...

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CHAPTER 7 INTEGRALS AND TRANSCENDENTAL FUNCTIONS 7.1 THE LOGARITHM DEFINED AS AN INTEGRAL 1. dx ln x ln 2 ln 3 ln 2. dx ln 3x 2 ln 2 ln 5 ln '' 3 1 2 0 " ±# ! ±$ ±" ±# x3 3 x 5 23 2 œ œ±œ œ ± cd c d kk k k 3. dy ln y 25 C 4. dr ln 4r 5 C 2y y2 5 4r 5 8r # # ± ± # # œ± ² œ ± ² 5. Let u 6 3 tan t du 3 sec t dt; œ² Ê œ # dt ln u C ln 6 3 tan t C 3 sec t du 63 t a n t u # ² œœ ² ² k k 6. Let u 2 sec y du sec y tan y dy; dy ln u C ln 2 sec y C sec y tan y sec y u du ² ² k k 7. ; let u 1 x du dx; ln u C ' dx dx dx du 2x2 x 2x1 x x x u ÈÈ È È È È ˆ‰ ²² # ² " ² Ê œ œ œ ² È ' ln 1 x C ln 1 x C œ²² ¸¸ 8. Let u sec x tan x du sec x tan x sec x dx (sec x)(tan x sec x) dx sec x dx ; Ê œ ² œ ² Ê œ ab # du u (ln u) du 2(ln u) C 2 ln (sec x tan x) C ' sec x dx du ln (sec x tan x) ul n u u ² ±"Î# "Î# " œ ² œ ² ² ' ' È 9. e dx e e e 3 2 1 10. e dx e e e 1 2 1 ln 2 ln 2 ln 3 ln 3 x x ln 3 ln 2 x x ln 2 ln 3 0 ln 2 ln 2 œ ± œ œ ± œ ± ²œ ± ² œ c d ± ±± ! ± 11. 8e dx 8e C 12. 2e dx e C Ð²Ñ Ð ±Ñ x1 2 2 13. e dx 2e 2 e e 2 e e 2(3 2) 2 ' ln 4 ln 9 x 2 x 2 ln 9 2 ln 4) 2 ln 3 ln 2 ln 9 ln 4 ÎÎ Ð Ñ Î Ð Î ± œ ± œ ± œ ±‘ ± ‘ˆ 14. e dx 4e 4 e e 4 e 1 4(2 1) 4 ' 0 ln 16 x4 l n1 6 4 0 l n2 ln 16 0 Ð Ñ Î ± œ ± œ ± œ ˆ ‰ˆ 15. Let u r du r dr 2 du r dr; œÊ œ Ê œ "Î# ±"Î# ±"Î# " # dr e r dr 2 e du 2e C 2e C 2e C e r ru u r r È r È È œ ² œ ² œ ² ' "Î# "Î# ±"Î# 16. Let u r du r dr 2 du r dr; Ê Ê ± œ "Î# ±"Î# ±"Î# " # dr e r dr 2 2e C 2e C ' e r r r "Î# "Î# È r È È ± œ ± ² œ ± ² ± ±"Î# ± ± 17. Let u t du 2t dt du 2t dt; œ± Ê Ê ± œ # 2te dt e C e C tu u t ## œ± ² œ± ²
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464 Chapter 7 Integrals and Transcendental Functions 18. Let u t du 4t dt du t dt; œÊ œ Ê œ %$ $ " 4 t e dt e du e C '' $ "" tu t 44 %% œœ ± 19. Let u du dx du dx; ² Ê ²œ " xx x ## dx e C e C e x uu 1 x 1x Î # œ² œ ²±œ ² ± Î 20. Let u x du 2x dx du x dx; Ê œ Ê œ ±# ±$ ±$ " # dx e x dx e C e C e C ' e x xu u x 1 x Î # $ # # # œ ± œ ± œ ± ±± $ ± ± Î """ " ### # 21. Let u sec t du sec t tan t dt sec t tan t dt; œÊ œ Ê œ 1 1 11 du 1 e sec ( t) tan ( t) dt C C sec t u ee ÐÑ " 1 1 ± œ ± us e c t ab 1 22. Let u csc ( t) du csc ( t) cot ( t) dt; œ± Ê œ ²± ± 1 e csc ( t) cot ( t) dt e C e C csc t u u csc t Ð²Ñ 1 1 ± ± œ² ± œ² ± 23. Let u e du e dv 2 du 2e dv; v ln u , v ln u ; œÊœ Ê œ œ Ê œ œ Ê œ vv v 66 2e cos e dv 2 cos u du 2 sin u 2 sin sin 2 1 1 ln 6 6 ln 2 2 2 6 6 ÐÎÑ Î Î Î Î " 1 1 œ ² œ ² œ cd ±‘ ˆ ˆ‰ 24. Let u du 2xe dx; x 0 u 1, x ln u e ; œ Ê Ê œ œ l n È 1 2xe cos e dx cos u du sin u sin ( ) sin (1) sin (1) 0.84147 01 ln È 1 Š‹ œ œ œ ² ¸² " 1 25. Let u 1 du e dr; œ± Ê œ rr dr du ln u C ln 1 e C e 1e u r r r ² " ± œ ± ± kk a b 26. dx dx; " ²² e 1 e x œ let u e 1 du e dx du e dx; Êœ ² Ê ² œ ± x dx du ln u C ln e 1 C e e1 u x x x ² " ± ± œ² ± ± a b 27.
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This note was uploaded on 04/17/2008 for the course MA 113 taught by Professor Massman during the Spring '08 term at Rose-Hulman.

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thomasET_226348_ism39 - CHAPTER 7 INTEGRALS AND...

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