the Further Mathematics network

www.fmnetwork.org.uk

the Further Mathematics network

www.fmnetwork.org.uk

FP2 (MEI)Calculus 2: Integration

using the inverse trigonometric functions

Let Maths take you Further…

Integration using the inverse trigonometric functionsBefore 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).

arcsinx arctan xarccosx

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).

More functions that we can now integrate!!

Some standard results….

This result is in the exam formula book

Consider…for example…

Let

This result is in the exam formula book

Three possible methods:

Algebraic manipulation

Substitution

Trigonometric substitution

Algebraic manipulation

Substitution

Trigonometric substitution

Three possible methods:

Algebraic manipulation

Substitution

Trigonometric substitution

Harder integrations

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).

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.