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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.