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# Integrating Sin2xCosx watch

1. In the Solomon paper L for C4 (OCR), theres the following question:

Evaluate
(with limits b = pi/3, and a = 0)

∫ Sin2x.Cosx dx

In the markscheme they've solved it as:

∫2Sinx.Cos²x dx

Which i assume they got from expanding Sin2x = 2SinxCosx

[(-2/3).Cos³x] with the limits from above.

I assume what they've done here is use a substitution of u=Cosx, so du/dx = -Sinx, so the integral becomes:

∫(2Sinx.u²) / -sinx = ∫ -2u² du = [ (-2/3)u³ ] = [ (-2/3)Cos³x ]

However i confused myself here thinking it wasnt correct to integrate Cos²x like this, as i thought that when integrating cos²x and sin²x, i had to use the double angle identities.

So is what i've done here the correct way to solve it, even though i didnt think it was correct maths to integrate cos²x this way? And whats the best way for me to identify which integration method to use on this sort of question? (I thought i had to integrate this by parts when i first saw it).
2. You weren't integrating cos²x here -- it was cos²x.sinx, which is very different.

The markscheme used the "function and its derivative" rule, which is essentially, as you spotted, using the substitution u=cosx with du/dx=-sinx.

If you did have cos²x by itself, then you would've needed to use the appropriate double-angle identity for cos2x.
3. Indeed, spotting a function and derivative is handy for integrating. And a substitution in this case is fine. All you've got to practise is being able to spot such cases!
4. Ok, thanks for the help.
5. Alternatively,
Using sin(A)cos(B) = 0.5[ sin(A+B) + sin(A-B) ]
Sin(2x)Cos(x) = 0.5[ sin(3x) + sin(x) ]
Which integrates to
0.5[ -(1/3)cos(3x) - cos(x) ] + c
Limits:
0.5[ (1/3) - (1/2) + (1/3) + 1 ]
= 7/12
6. (Original post by Migraine)
∫2Sinx.Cos²x dx
Which i assume they got from expanding Sin2x = 2SinxCosx
[(-2/3).Cos³x] with the limits from above.
its done my a method called "recognition" - you merely look at the integral and work backwards saying "what function must i differentiate wto get the integral"

wol ah,

regarding method - what tehy tell you to use - if its an easy one, they'll explect you dto do it..ones like that they may well give you a substitution because the majority of people cant do recognition

pk
7. Is that the same as "standard patterns"?
8. (Original post by seankhn)
Is that the same as "standard patterns"?
You what
9. I would say it's done by knowing the chain rule that g(f(x)) differentiates to

g'(f(x)) f'(x)

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