Integration

1. Integration

2. Esme • Esme is approaching her last topic in her Al level mathematics programme. She goes for a walk through the forest to reflect on her two years of study and for her love of mathematics. Esme stumbles across a very special tree.

3. The Integration Methods Tree • Standard Integrals • Inverse of Chain rule • Using Trig Identities • Partial fractions • Substitution • Integration by Parts

4. A little later Integration Applications • Finding the area using integration and trapezium rule • Finding the volumes of solids of revolution

5. Branch 1 Standard Integrals • Match the integral with the correct answers.

10. Branch 2 Reverse of the Chain Rule • Instead of putting the power in front and dropping the power by one and multiplying by the derivative of the bit in the middle • Y = (2x+4)6 • Y’ = 6(2x+4)5X2 • We raise the power by one, divide by that power and divide by the derivative of the bit in the middle

11. The Magic Phrase • We raise the power by one, divide by that power and divide by the derivative of the bit in the middle

12. Example

13. Branch 3 Using trig identities to Integrate • Do you remember all our trig identities from C3? • Write down as many as you can remember

14. Example 1

15. More Examples

16. Example

19. Branch 4 Using partial fractions • We can use partial fractions to integrate expressions that are too long to do other methods with! • What are those rules again?

21. Ex 6D Question 1e

22. Examples

23. Improper Fraction

25. More Standard Integrals Recognizing

26. Recognizing the Reverse of the Chain Rule

27. Recognizing Ln and reverse of Chain rule! • Here remember :

28. Examples

29. Example

30. Another example

31. Your example

32. Your example

33. A volunteer

34. A volunteer

35. Branch 5 Integration using Substitution • Here is another post it lesson! • Sometimes it is much easier to substitute a simpler function that we can easily integrate • Integrate the following using the substitution provided

36. Example 1 Remember to substitute back and c!

37. Example 3

38. Example 3

39. Definite Integrals • Here remember to change your boundaries using the substitution you have used • Let’s look at the first example but this time we want to evaluate this integral from x = 1 to x= 3

40. Example 4 Here you can use the new boundaries 7 and 11! No need to add c or substitute back! Why?

43. Branch 6 Using Integration by Parts • Do you remember the product rule to differentiate the product of two functions? • Or in words v du + u dv • We can actually rearrange this to help us to integrate two functions multiplied together!

44. Integration by parts Rearranging Integrating

45. Integration by parts how do I use this?

46. Example 2

47. Example 3 with boundaries

49. Five review questions

50. Five review questions