1. By Zac Cockman Liz Mooney
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
n cannot 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
U=x
Du=dx
N=2
N=1
N=0

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

11. Try Me

12. Try Me
U = x4 + 3
Du = 4x3 dx
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
U = cosx
Du = -sinx
N = 4
N = 6
N = 8
-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
U = cosx
Du = -2sin2x
N = 2
N = 4
-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
U = cosx
du= -sinxdx
n = 3
n = 5

30. Powers of Trig Odd-Odd
Example

31. Powers of Trig Odd Odd Example Continued U = cosx Du = -sinx N = 17

32. Powers of Trig Even-Even
Change to half angle formula
∫sin2xdx
∫1-cos2xdx 2
1/2∫dx-(1/2)(1/2)∫2cos2xdx
U = x
U = 2x
Du = dx
Du = 2dx
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 Nostrand Reinnhold Company Inc., 1976