How do I integrate this? (C3 reverse chain rule)

How do I integrate (cosxsin2x)^1/2?

I use the Guess Differentiate Adjust method, but so far I have no idea how to start. Can someone please help me out?

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Reply 1

username1732133

Original post by Mystelle

How do I integrate (cosxsin2x)^1/2?

I use the Guess Differentiate Adjust method, but so far I have no idea how to start. Can someone please help me out?

A first step.

\sin 2x \equiv 2\sin x \cos x

Reply 2

black1blade

It's definitely nothing to do with reverse chain rule as that's only on functions of linear functions of x.

Reply 3

MiladA

Use the identity sin2x=2sinxcosx, simplify, then make a substitution.

Reply 4

frostfly

Original post by MiladAhmed

Use the identity sin2x=2sinxcosx, simplify, then make a substitution.

what do you mean by make a substitution?

Reply 5

Aph

Original post by black1blade

It's definitely nothing to do with reverse chain rule as that's only on functions of linear functions of x.

Not true, the chain rule, in its simplist form states

\frac{dy}{dx} f(g(x)) = g'(x)*f'(g(x))

For any functions g(x) or f(x)

Reply 6

Notnek

Original post by Aph

Not true, the chain rule, in its simplist form states

\frac{dy}{dx} f(g(x)) = g'(x)*f'(g(x))

For any functions g(x) or f(x)

It depends how you interpret "reverse chain rule". Often when textbooks refer to "reverse chain rule". they mean integration of functions of linear functions.

Reply 7

black1blade

Original post by Aph

Not true, the chain rule, in its simplist form states

\frac{dy}{dx} f(g(x)) = g'(x)*f'(g(x))

For any functions g(x) or f(x)

But when you are integrating if you have a function g(f(x)) then f(x) must be linear to use the "reverse chain rule" trick of integrating g(f(x)) with respect to f(x) and dividing by f'(x). With that method you can integrate sin(2x+4) but not sin(cos(x)).

Reply 8

black1blade

Original post by Mystelle

what do you mean by make a substitution?

Really this is a core 4 problem.

Reply 9

Aph

Original post by Notnek

It depends how you interpret "reverse chain rule". Often when textbooks refer to "reverse chain rule". they mean integration of functions of linear functions.

I've forgotten what a-level maths is like :lol:

Original post by black1blade

The derivative of sin(cos(x)) is -sin(x)cos(cos(x)), you should be able to recognise that as a chain rule derivative and then intergrate it.

Reply 10

SerBronn

The solution does rely on recognizing an expression of the form f(x)^(1/2) f'(x) so I think it's reasonable to describe it as an example of reverse chain rule. @Mystelle, two people have already told you the first step to do.

Reply 11

Notnek

Original post by black1blade

The other interpretation of "reverse chain rule" is to actually reverse the chain rule :smile:

\int 2xe^{x^2} \ dx = e^{x^2} + c

may be considered "reverse chain rule" since it reverses the chain rule differentiation of

e^{x^2}

, without the need for a substitution. But this may be called "integration using standard patterns" or recognition or something like that.

Textbooks cause this confusion and teachers usually follow the textbooks.

Reply 12

black1blade

Original post by Aph

I've forgotten what a-level maths is like :lol:

The derivative of sin(cos(x)) is -sin(x)cos(cos(x)), you should be able to recognise that as a chain rule derivative and then intergrate it.

Yeah of course you can differentiate sin(cosx) but you can't integrate it with core 3 knowledge.

Reply 13

SerBronn

Original post by Notnek

The other interpretation of "reverse chain rule" is to actually reverse the chain rule :smile:

\int 2xe^{x^2} \ dx = e^{x^2} + c

may be considered "reverse chain rule" since it reverses the chain rule differentiation of

e^{x^2}

Thank you for clarifying this as I was puzzled why black1blade thought the (reverse) chain rule only applied to linear functions. It's not his fault if he's been taught that nomenclature. These textbook authors should probably be shot though.

Reply 14

black1blade

Original post by SerBronn

Yeah we don't have to work out substitutions either so ngl struggling to find the sub on this one XD.

Reply 15

DFranklin

Original post by Notnek

Depends on the exam board.

Am I missing something here? I don't see any way to integrate sin(cos(x))...

Reply 16

black1blade

Original post by DFranklin

Am I missing something here? I don't see any way to integrate sin(cos(x))...

I don't think he was talking about specific example but in some exam boards I guess they do integration by substitution in c3.

Reply 17

Notnek

Original post by DFranklin

Am I missing something here? I don't see any way to integrate sin(cos(x))...

Sorry I was getting mixed up with different posts. I thought black1blade was saying that integration by substitution is not part of C3.

Reply 18

Notnek

Original post by Mystelle

what do you mean by make a substitution?

This thread is getting messy - sorry about that.

Can you please clarify which exam board you are doing and what methods you have learnt in C3 e.g. have you learnt integration by substitution?

And can you please post all your working up to the point you get stuck.