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C4 Integration

I think integration is a decent topic but I have no idea how to integrate the following:

•

4Cos3xSin2x

•

Sec2xtan2x




Guides on how to integrate these would be greatly appreciated,

Thanks
Original post by Leking9
I think integration is a decent topic but I have no idea how to integrate the following:

•

4Cos3xSin2x

•

Sec2xtan2x




Guides on how to integrate these would be greatly appreciated,

Thanks


Do you know how to integrate CosxSinnx
Original post by Leking9
I think integration is a decent topic but I have no idea how to integrate the following:

•

4Cos3xSin2x

•

Sec2xtan2x




Guides on how to integrate these would be greatly appreciated,

Thanks


For the second one recall what the differential of Tanx is
Reply 3
Avatar for Leking9
Leking9
OP
11
Original post by TenOfThem
Do you know how to integrate CosxSinnx


Not really, the only way I know is to do it by parts but that's extremely long
Reply 4
Avatar for Leking9
Leking9
OP
11
Original post by TenOfThem
For the second one recall what the differential of Tanx is



sec2​x
Original post by Leking9
Not really, the only way I know is to do it by parts but that's extremely long



Original post by Leking9
sec2​x



Have you done inverse chain rule .... If so these are both easy

If not substitution ... U=sinx and u=tanx
Reply 6
Avatar for Leking9
Leking9
OP
11
Original post by TenOfThem
Have you done inverse chain rule .... If so these are both easy

If not substitution ... U=sinx and u=tanx



"Inverse chain rule" I've been taught according to what's inside the C4 edexcel book and for some reason I can't see any exercises on it nor has my teacher gone over it
Original post by Leking9
"Inverse chain rule" I've been taught according to what's inside the C4 edexcel book and for some reason I can't see any exercises on it nor has my teacher gone over it


It is not called that in the book

It is just by recognition

If you know what sinx differentiates to, then you know what (sinx)^n differentiates to

Therefore you can integrate cosx (sinx)^n



Or use the substitutions that I pointed out
Reply 8
Avatar for TVIO
TVIO
12
Original post by Leking9
"Inverse chain rule" I've been taught according to what's inside the C4 edexcel book and for some reason I can't see any exercises on it nor has my teacher gone over it


It's called integration by parts and you definitely should have covered it?
Original post by TVIO
It's called integration by parts and you definitely should have covered it?


These questions do not require IBP
Reply 10
Avatar for Phichi
Phichi
11
Original post by Leking9
"Inverse chain rule" I've been taught according to what's inside the C4 edexcel book and for some reason I can't see any exercises on it nor has my teacher gone over it


It's noting that the form of the integral is in f′(x)f(x)f'(x)f(x), and answering the question via recognition. In reality it's just a simple substitution, but with time you'll be able to do them by recognition.

∫cos(x)sinn(x) dx\displaystyle \int cos(x)sin^n(x) \, dx

u=sin(x)   ⇒  dudx=cos(x)   ⇒  dx=ducos(x)u = sin(x) \, \, \, \Rightarrow \, \, \dfrac{du}{dx} = cos(x) \, \, \, \Rightarrow \, \, dx = \dfrac{du}{cos(x)}

Thus the integral becomes:

∫cos(x)uncos(x) du   ⇒  ∫un du=un+1n+1+c=sinn+1(x)n+1+c\displaystyle \int \dfrac{cos(x)u^n}{cos(x)} \, du \, \, \, \Rightarrow \, \, \int u^n \, du = \dfrac{u^{n+1}}{n+1} + c = \dfrac{sin^{n+1}(x)}{n+1} + c
Original post by TenOfThem
These questions do not require IBP


could use sin(a+-b) formulas


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(edited 9 years ago)
Original post by physicsmaths
could use sin(a+-b) formulas


Posted from TSR Mobile


Could you?

Since both of these integrals are very straightforward I would be wary of introducing anything new
Original post by TenOfThem
Could you?

Since both of these integrals are very straightforward I would be wary of introducing anything new


That's how I always done the sinnxcosmx ones. It's just using the trig formulas to cancel into a very simple integral.


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4cos(3x)sin(2x)= 2sin(5x) -2sinx


Posted from TSR Mobile
Original post by physicsmaths
That's how I always done the sinnxcosmx ones. It's just using the trig formulas to cancel into a very simple integral.


Posted from TSR Mobile


Those questions .... Yes

Not the ones in the OP though
Original post by TenOfThem
Those questions .... Yes

Not the ones in the OP though


Oh, is that raised to the power 3? I thought it was cos(3x)sin(2x)
My bad...


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