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Integrals

Integrals. By Zac Cockman Liz Mooney. Integration Techniques. Integration is the process of finding an indefinite or diefinite integral Integral is the definite integral is the fundamental concept of the integral calculus. It is written as

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Integrals

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  1. Integrals By Zac Cockman Liz Mooney

  2. Integration Techniques • Integration is the process of finding an indefinite or diefinite integral • Integral is the definite integral is the fundamental concept of the integral calculus. It is written as • Where f(x) is the integrand, a and b are the lower and upper limits of integration, and x is the variable of integration.

  3. Integration techniques • Integration is the opposite of Differentiation. • Power Rule • U-Substitution • Special Cases • Sin and Cos

  4. Power Rule • ncannot equal -1 • u=x • Du=dx • N=1 • C = constant • + c

  5. Examples U = 2x Du = dx n = 1 Answer = X2 + C

  6. Examples • Answer = X3/3 + 3x2/2 + 2x + C

  7. Examples • U = 4 X2 • Du = 8xdx • N = -1/2 • 3/8 * 2 * (4x2 + 5)1/2 + C • Answer ¾(4X2 + 5)1/2 + C

  8. Try Me

  9. Try Me

  10. Try Me Continue • U = 1 +x2 • Du = 2dx • N = -1/2 • 1/2 [2(1+x2)1/2] + C • (1+x2)1/2 + C

  11. Try Me

  12. Try Me • U = x4 + 3 • Du = 4x3dx • N = 2

  13. Try Me Continued

  14. U-Sub • What is U-Sub • When do you use it • Steps • Find your u, du, and for u, solve for x • Replace all the x for u. • Do the same steps for power rule • At the end replace the u in the problem for your u when you found it in the beginning.

  15. Example • U= • X= u2 -1 • dx= 2udu • (u2 – 1) u(2udu) • 2u4 – 2u2 • 2/5 (u5 – 2/3u3) + c • 2/5 (x+1) 5/2 + c

  16. Example • U = • U2 – 1 = x • 2udu = dx 2/3(x+1)3/2 -2(x+1) + c

  17. Try Me

  18. Try Me U = X = Du = udu 1/10 u5 + 1/2u2 + c 1/10 (2x-3)5/2 + ½(2x-3) + c

  19. Special Cases • When n = -1 the u is put inside the absolute value of the natural log • If there is only one x in the problem and it is squared, square the term before taking the interval

  20. Special Cases • Examples • U = x-1 • Du = dx • N = -1

  21. Special Cases • Examples

  22. Integration using Powers of Sin and Cos • Three Methods • Odd-Even Odd-Odd Even-Even • In Odd-Even, take the odd power and re write the odd power as odd even • Re write the even power change it using Pythagorean identity. • In Odd-Odd, take one of the odds, change to odd even • Use same rules

  23. Integration using Powers of sin and cos • For Even-Even, change the power to the half angle formula. Special Case If the Power of the trig is 1, u is the angle

  24. Powers of Trig Odd - Even • Take the odd power, re write the odd power as odd even • Re write the even power, change it using the Pythagorean identities. • ∫sin5xcos4xdx • ∫sin4x sinxcos4xdx • ∫(1-cos2x)2 sinxcos2xdx

  25. Powers of Trig Odd-Even • ∫(1-2cos2x+cos4x) sinxcos4xdx • ∫sinxcos4xdx-2 ∫sinx cos6xdx+∫sinxcos8xdx -1/5cos5x+2/7cos7x-1/9cos9x+c

  26. Try Me ∫sin32xcos22xdx

  27. Powers of Trig Odd - Even • Try Me • ∫sin32xcos22xdx • ∫sin22xsin2xcos22xdx • ∫(1-cos22x) sin2xcos22xdx • -1/2 ∫sin22xcos22xdx+1/2 ∫sin2xcos42xdx -1/6 cos32x+1/10cos52x+c

  28. Powers of Trig Odd Odd • Take one of the odds, change to odd even. Use other rules to finish. • Example

  29. Powers of trig Odd-Odd

  30. Powers of Trig Odd-Odd • Example

  31. Powers of Trig Odd Odd • Example Continued

  32. Powers of Trig Even-Even • Change to half angle formula • ∫sin2xdx • ∫1-cos2xdx • 2 • 1/2∫dx-(1/2)(1/2)∫2cos2xdx 1/2x-1/4sin2x+c

  33. Try Me ∫sin2xcos2x

  34. Powers of Trig Even-Even • Try Me • ∫sin2xcos2x • ∫(1-cos2x)(1+cos2x)        2            2 • 1/4∫(1-cos22x)dx • 1/4∫sin22xdx • 1/4∫1-cos4x/2dx • 1/8∫dx-(1/4)(1/8) ∫4cos4xdx • 1/8x-1/32sin4x+c

  35. Solving for Integrals • U =x-1 • Du = dx • N = 2 • 9 – 0 = 9

  36. Try Me • Try Me

  37. Solving for Integrals • U = x2 + 2 • Du = 2xdx • N = 2

  38. Bibliography • www.musopen.com • Mathematics Dictionary, Fourth Edition, James/James, Van NostrandReinnhold Company Inc., 1976

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