Integration I KUS objectives BAT Use standard Trig

Integration I • KUS objectives BAT Use standard Trig identities to integrate functions Starter: copy and complete the identities

WB 1 a integrate standard functions You met the following in differentiation:

WB 1 b The modulus sign is used here to avoid potential problems with negative numbers… (More info on the next slide!) Therefore, you already can deduce the following

So far we have these ‘rules’ 1) Integrate the function using what you know from differentiation 2) Divide by the coefficient of x 3) Simplify if possible and add C These apply ALL the functions f(x) you know

Consider starting with sin(2 x + 3) and the answer that would give This is double what we are wanting to integrate Therefore, we must ‘start’ with half the amount… This is a VERY common method of integration – considering what we might start with that would differentiate to our answer…

Consider starting with tan 3 x and the answer that would give This is three times what we are wanting to integrate Therefore, we must ‘start’ with a third of the amount…

Write as many Trig Identities as you can from memory

WB 4 a Find the following integral: Expand the bracket Replace tan 2 x Simplify Integrate separately

WB 4 b Integrate the following: In this case consider the power of sine. If it has been differentiated, it must have been sin 3 x originally… Write as a cubed bracket Differentiate using the chain rule Rewrite – this has given us exactly what we wanted! Don’t forget the + C!

WB 4 c Integrate the following: Write using powers Imagine how we could end up with a -3 as a power. . . Use the chain rule Rewrite This is double what we want so multiply the ‘guess’ by 1/2

WB 4 d Integrate the following: Consider using a power 5 Write as a bracket to the power 5 Differentiate using the chain rule We have an extra secx HOWEVER: We cannot just add this to our ‘guess’ as before, as the differentiation will need to be performed using the product rule , rather than the Chain rule! We need to find another way!

WB 4 d (cont) Integrate the following: Consider using a power 4 Write as a bracket to the power 4 Differentiate using the chain rule This is what we want, but 4/5 of the amount Multiply by the guess by 5/4

We can also use the formulas from the formula booklet

The formula booklet gives you some help We need more techniques before meeting all of these

KUS objectives BAT Use standard Trig identities to integrate functions self-assess One thing learned is – One thing to improve is –

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