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

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    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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    (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.
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    It's definitely nothing to do with reverse chain rule as that's only on functions of linear functions of x.
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    Use the identity sin2x=2sinxcosx, simplify, then make a substitution.
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    (Original post by MiladAhmed)
    Use the identity sin2x=2sinxcosx, simplify, then make a substitution.
    what do you mean by make a substitution?
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    (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)
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    (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.
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    (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)).
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    (Original post by Mystelle)
    what do you mean by make a substitution?
    Really this is a core 4 problem.
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    (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)
    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)).
    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.
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    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.
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    (Original post by black1blade)
    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)).
    The other interpretation of "reverse chain rule" is to actually reverse the chain rule

    So \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.
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    (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.
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    (Original post by Notnek)
    The other interpretation of "reverse chain rule" is to actually reverse the chain rule

    So \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.
    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.
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    (Original post by SerBronn)
    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.
    Yeah we don't have to work out substitutions either so ngl struggling to find the sub on this one XD.
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    (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))...
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    (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.
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    (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.
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    (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.
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    (Original post by Notnek)
    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.
    We've been taught the Guess, Differentiate, Adjust method. Here's an example of another question I did.

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    So far I've only managed to write sin2x as 2sinxcosx, as suggested in the replies.
 
 
 
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