Calculus: One and Several Variables

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Unformatted text preview: P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-08 JWDD027-Salas-v1 December 4, 2006 20:43 404 SECTION 8.1 CHAPTER 8 SECTION 8.1 1. e 2 − x dx = − e 2 − x + C 2. cos 2 3 xdx = 3 2 sin 2 3 x + C 3. 1 sin πxdx = − 1 π cos πx 1 = 2 π 4. t sec πx tan πxdx = 1 π sec πx t = 1 π (sec πt − 1) 5. sec 2 (1 − x ) dx = − tan(1 − x ) + C 6. dx 5 x = 5 − x dx = − 1 ln5 5 − x + C = − 1 5 x ln5 + C 7. π/ 3 π/ 6 cot xdx = ln(sin x ) π/ 3 π/ 6 = ln √ 3 2 − ln 1 2 = 1 2 ln 3 8. 1 x 3 1 + x 4 dx = 1 4 ln(1 + x 4 ) 1 = 1 4 ln2 9. u = 1 − x 2 du = − 2 xdx ; xdx √ 1 − x 2 = − 1 2 u − 1 / 2 du = − u 1 / 2 + C = − 1 − x 2 + C 10. π/ 4 − π/ 4 dx cos 2 x = π/ 4 − π/ 4 sec 2 xdx = tan x π/ 4 − π/ 4 = 2 11. π/ 4 − π/ 4 sin x cos 2 x dx = π/ 4 − π/ 4 sec x tan xdx = sec x π/ 4 − π/ 4 = 0 12. e √ x √ x dx = 2 e √ x + C 13. u = 1 /x du = dx/x 2 x = 1 ⇒ u = 1 x = 2 ⇒ u = 1 / 2 ; 2 1 e 1 /x x 2 dx = 1 / 2 1 − e u du = − e u 1 / 2 1 = e − √ e 14. x 3 √ 1 − x 4 dx = − 1 4 du √ u = − 1 2 √ u + C = − 1 2 1 − x 4 + C 15. c dx x 2 + c 2 = 1 c arctan x c c = π 4 c 16. a x e x dx = ( ae ) x dx = ( ae ) x ln( ae ) + C = a x e x 1 + ln a + C P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-08 JWDD027-Salas-v1 December 4, 2006 20:43 SECTION 8.1 405 17. u = 3 tan θ + 1 du = 3 sec 2 θ dθ ; sec 2 θ √ 3 tan θ + 1 dθ = 1 3 u − 1 / 2 du = 2 3 u 1 / 2 + C = 2 3 √ 3 tan θ + 1 + C 18. sin φ 3 − 2cos φ dφ = 1 2 du u = 1 2 ln | u | + C = 1 2 ln(3 − 2cos φ ) + C 19. e x ae x − b dx = 1 a ln | ae x − b | + C 20. dx x 2 − 4 x + 13 = dx ( x − 2) 2 + 9 = 1 3 arctan x − 2 3 + C 21. u = x + 1 du = dx ; x ( x + 1) 2 + 4 dx = u − 1 u 2 + 4 du = u u 2 + 4 du − du u 2 + 4 = 1 2 ln | u 2 + 4 | − 1 2 arctan u 2 + C = 1 2 ln | ( x + 1) 2 + 4 | − 1 2 arctan x + 1 2 + C 22. ln x x dx = 1 2 (ln x ) 2 + C 23. u = x 2 du = 2 xdx ; x √ 1 − x 4 dx = 1 2 du √ 1 − u 2 = 1 2 arcsin u + C = 1 2 arcsin( x 2 ) + C 24. e x 1 + e 2 x dx = du 1 + u 2 = arctan u + C = arctan e x + C 25. u = x + 3 du = dx ; dx x 2 + 6 x + 10 = dx ( x + 3) 2 + 1 = du u 2 + 1 = arctan u + C = arctan( x + 3) + C 26. e x tan e x dx = tan udu = ln | sec u | + C = ln | sec e x | + C 27. x sin x 2 dx = − 1 2 cos x 2 + C 28. x + 1 √ 1 − x 2 dx = x √ 1 − x 2 dx + dx √ 1 − x 2 = − 1 − x 2 + arcsin x + C 29. tan 2 xdx = (sec 2 x − 1) dx = tan x − x + C P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU JWDD027-08 JWDD027-Salas-v1 December 4, 2006 20:43 406 SECTION 8.1 30. cosh2 x sinh 3 2 xdx = 1 8 sinh 4 2 x + C 31. u = ln x du = dx/x x = 1 ⇒ u = 0 x = e ⇒ u = 1 ; e 1 ln x 3 x dx = e 1 3 ln x x dx = 3 1 udu = 3 u 2 2 1 = 3 2 32....
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This note was uploaded on 04/30/2011 for the course MATH 1431 taught by Professor Any during the Spring '08 term at University of Houston.

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ch08[1] - P1: PBU/OVY P2: PBU/OVY QC: PBU/OVY T1: PBU...

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