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    Hi,

    I understand how to get to the answer, that's fine. But I just don't understand the logic used. "Consider", what do they mean? Are we just "Letting" y=... or is y=sin(2x+3) the actual integral?

    If so, if we differentiate this value, we should get back to cos(2x+3) but we clearly don't. So this value cannot be the integral? Why are we taking these steps?

    Thanks!
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    (Original post by ps1265A)


    Hi,

    I understand how to get to the answer, that's fine. But I just don't understand the logic used. "Consider", what do they mean? Are we just "Letting" y=... or is y=sin(2x+3) the actual integral?

    If so, if we differentiate this value, we should get back to cos(2x+3) but we clearly don't. So this value cannot be the integral? Why are we taking these steps?

    Thanks!
    The answer is y=1/2 sin(2x+3)

    If you differentiate that you get cos(2x+3)

    The text is explaining why you you need the 1/2
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    (Original post by TenOfThem)
    The answer is y=1/2 sin(2x+3)

    If you differentiate that you get cos(2x+3)

    The text is explaining why you you need the 1/2
    I want to know whether step 1 is just a "rough" integration


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    (Original post by ps1265A)
    I want to know whether step 1 is just a "rough" integration


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    No

    Step 1 is telling you the thinking process
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    (Original post by TenOfThem)
    No

    Step 1 is telling you the thinking process
    And what is the thinking process? An estimation?


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    (Original post by ps1265A)
    And what is the thinking process? An estimation?


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    It is looking at the integrated and saying to yourself ... Oh ... I must have differentiated a Sin function in order to get a Cos function


    I am struggling to see what is confusing you
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    (Original post by ps1265A)
    I want to know whether step 1 is just a "rough" integration


    Posted from TSR Mobile
    Step 1 is an "intelligent " guess. You then check whether it is correct and adjust if necessary.
    You should try to get to know certain standard patterns. For example, consider a function of x f(x). If the integral is F(x) then the integral of f(ax) will be F(ax)/a
    Similarly the integral of f(ax+b) will be F(ax+b)/a.
    IN reverse, if the derivative of f(x) is g(x) say then the derivative of f(ax) will be ag(ax) and of f(ax+b) will be ag(ax+b).
    More generally the derivative of f(g(x)) will be g'(x)f'(g(x))
    See page 38 in the attachment.
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  1. File Type: pdf Notes for C1-C4.pdf (918.2 KB, 501 views)
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    (Original post by TenOfThem)
    It is looking at the integrated and saying to yourself ... Oh ... I must have differentiated a Sin function in order to get a Cos function


    I am struggling to see what is confusing you
    Yes, that's exactly what I wanted to know and was thinking the same, just didn't know how to word it


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    (Original post by brianeverit)
    Step 1 is an "intelligent " guess. You then check whether it is correct and adjust if necessary.
    You should try to get to know certain standard patterns. For example, consider a function of x f(x). If the integral is F(x) then the integral of f(ax) will be F(ax)/a
    Similarly the integral of f(ax+b) will be F(ax+b)/a.
    IN reverse, if the derivative of f(x) is g(x) say then the derivative of f(ax) will be ag(ax) and of f(ax+b) will be ag(ax+b).
    More generally the derivative of f(g(x)) will be g'(x)f'(g(x))
    See page 38 in the attachment.
    Thank you soooo much, I understand it now


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    (Original post by ps1265A)
    Thank you soooo much, I understand it now


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    You''re welcome
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    (Original post by brianeverit)
    You''re welcome
    Could you tell me what the best way to learn integration is? At the moment, I have 2 option: one is by simply looking at the formulas and just plugging in and one is by actually doing the steps as I did above. Or do I have to adopt a combination of both because all questions don't follow the above method?


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    (Original post by TenOfThem)
    It is looking at the integrated and saying to yourself ... Oh ... I must have differentiated a Sin function in order to get a Cos function


    I am struggling to see what is confusing you
    Could you tell me what the best way to learn integration is? At the moment, I have 2 option: one is by simply looking at the formulas and just plugging in and one is by actually doing the steps as I did above. Or do I have to adopt a combination of both because all questions don't follow the above method?


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    (Original post by brianeverit)
    You''re welcome
    Because say if I have the Q was integrate 1/(2x+1)^2 and I followed the 3 step method, I would get:
    Name:  ImageUploadedByStudent Room1421517816.692358.jpg
Views: 103
Size:  141.5 KB

    But the actual answer is -1/2(2x+1)^2


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    (Original post by ps1265A)
    Because say if I have the Q was integrate 1/(2x+1)^2 and I followed the 3 step method, I would get:
    Name:  ImageUploadedByStudent Room1421517816.692358.jpg
Views: 103
Size:  141.5 KB

    But the actual answer is -1/2(2x+1)^2


    Posted from TSR Mobile
    Both this question and the original question use the idea of "reverse chain rule"

    You should know that differentiating the function of a function needs the chain rule

    So if you are integrating the function of a function you will consider reversing the chain rule
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    (Original post by TenOfThem)
    Both this question and the original question use the idea of "reverse chain rule"

    You should know that differentiating the function of a function needs the chain rule

    So if you are integrating the function of a function you will consider reversing the chain rule
    I have used the reverse chain rules


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    (Original post by ps1265A)
    I have used the reverse chain rules


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    Sorry did not look at your work properly

    You need to increase the power

    You have changed a power of -2 to a power of -3

    That is decreasing
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    (Original post by TenOfThem)
    Sorry did not look at your work properly

    You need to increase the power

    You have changed a power of -2 to a power of -3

    That is decreasing
    I'm sorry if the picture is unclear. I've changed my power from a 2 to a 3


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    (Original post by TenOfThem)
    Sorry did not look at your work properly

    You need to increase the power

    You have changed a power of -2 to a power of -3

    That is decreasing
    Or do I have to account for the reciprocal?


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    (Original post by ps1265A)
    Or do I have to account for the reciprocal?


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    I can see that you have changed from a 2to a 3

    As you have realised now ... You have changed -2 to -3 ... Decreasing rather than increasing ... Since the function is in the denominator
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    (Original post by TenOfThem)
    I can see that you have changed from a 2to a 3

    As you have realised now ... You have changed -2 to -3 ... Decreasing rather than increasing ... Since the function is in the denominator
    So this is what I've got:

    Name:  ImageUploadedByStudent Room1421520876.060318.jpg
Views: 91
Size:  142.9 KB
    Name:  ImageUploadedByStudent Room1421520911.005440.jpg
Views: 93
Size:  139.6 KB


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