Using our work from the last few weeks, work out the following integrals:1. cosx dx

2. sinx dx

3. cos3x dx

4. sin3x dx

5. sin2x dx

6. cos2x dxWhy are the last 2 difficultto answer?

Today

Using trig identities to help with difficult integrals

e.g. sin2x dx

tanx

1

sin2x + cos2x

sec2xtan2 x + 1

cosec2x

1 + cot2x

sinxcosx

Can you put these trig identities back together correctly?

sec2xtan2x + 1

cosec2x1 + cot2x

1sin2x + cos2x

tanx sinxcosx

Formulae we will be using today:

Proving these is beyond A Level, but can be worked out using the formula book

sin2x

cos2x

cos23x

sin25x

sinxcosx

sin3xcos3x

(cosx + 1)2

(cosx + sinx)2

sin2x ½(1- cos2x)

cos2x ½(cos2x + 1)

cos23x ½(cos6x + 1)

sin25x ½(1 – cos10x)

sinxcosx ½sin2x

sin3xcos3x ½sin6x

(cosx + 1)2 ½cos2x + 2cosx + 1½

(cosx + sinx)2 1 + sin2x

1. sin2x dx

2. cos2x dx

3. cos25x dx

4. sinxcosx dx

5. sin7xcos7x dx

6. (cosx + sinx)2 dx

Using identities to help, integratethe above.

0

π

Extension

The region enclosed by the curve y = cosx and the x-axis between x = 0 and x = π/2 is rotated through 2π radians about the x – axis. Show that the volume of the solid of revolution formed is π2/4

We met these formulae last week, they

helped us to integrate things like sin2x

and sinxcosx.

Today we are going to look at where they come from and how we canwork them out using the formula book.

Have a look at page 5of the formula book

The addition formulae and double angle formulae are helpful for integration, and also for solving equations and for finding minimums and maximums on graphs.