1. the Further Mathematics network www.fmnetwork.org.uk

2. the Further Mathematics network www.fmnetwork.org.uk FP2 (MEI) Calculus 2: Integration using the inverse trigonometric functions Let Maths take you Further…

3. Integration using the inverse trigonometric functions Before you start… You need to be familiar with the laws of indices (Core 1 chapter 5) and logarithms (Core 2 chapter 11). You need to be confident with all the techniques of differentiation and integration in C2 – C4, in particular differentiation using the chain rule, differentiation of trigonometric functions and implicit differentiation (C3 chapter 4), integration by substitution and integration of trigonometric functions (C3 chapter 5) and integration using partial fractions (C4 chapter 10). You must also be confident with all the work on Trigonometry covered so far, (C2 chapter 10 and C4 chapter 8). When you have finished… You should: Recognise integrals of functions of the form and be able to integrate associated functions by using trigonometrical substitutions (pages 11 – 14).

4. arcsinx arctan x arccosx

5. Summary of results (these are given in the exam formula book) Now that we have these results we can use the chain rule to differentiate composite functions that include inverse trigonometric functions (this was done in the previous tutorial).

9. More functions that we can now integrate!!

10. Some standard results…. This result is in the exam formula book Consider…for example…

11. Let

12. This result is in the exam formula book

13. Three possible methods: Algebraic manipulation Substitution Trigonometric substitution

14. Algebraic manipulation

15. Substitution

16. Trigonometric substitution

17. Three possible methods: Algebraic manipulation Substitution Trigonometric substitution

19. Harder integrations

29. Integration using the inverse trigonometric functions Now you have finished… Recognise integrals of functions of the form and be able to integrate associated functions by using trigonometrical substitutions (see further examples on pages 11 – 14).

30. Independent study: Using the MEI online resources complete the study plan for Calculus 2: Integration using the inverse trigonometric functions Do the online multiple choice test for this and submit your answers online.