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Maths C4 - Integration... Help??? Watch

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    So I've finally embarked onto the last chapter of C4 from the Edexcel Modular Textbook and already I am feeling quite overwhelmed! The reason being...
    Name:  C4 Ch.6 Integrating Standard Functions.png
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    Obviously I know I have technically covered all these as a lot of them are just the reverse process of what I learnt from the C3 Differentiation chapter. But even then I wasn't completely confident and feel like I will struggle to remember everything
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    (Original post by Philip-flop)
    So I've finally embarked onto the last chapter of C4 from the Edexcel Modular Textbook and already I am feeling quite overwhelmed! The reason being...
    Name:  C4 Ch.6 Integrating Standard Functions.png
Views: 213
Size:  74.6 KB

    Obviously I know I have technically covered all these as a lot of them are just the reverse process of what I learnt from the C3 Differentiation chapter. But even then I wasn't completely confident and feel like I will struggle to remember everything
    Practice makes perfect

    And also confirming which ones will be in your formula booklet.
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    (Original post by Philip-flop)
    So I've finally embarked onto the last chapter of C4 from the Edexcel Modular Textbook and already I am feeling quite overwhelmed! The reason being...


    Obviously I know I have technically covered all these as a lot of them are just the reverse process of what I learnt from the C3 Differentiation chapter. But even then I wasn't completely confident and feel like I will struggle to remember everything
    Check the Edexcel formula book (can Google it) and see which integrals are not there. These will be the ones you need to remember. Hopefully that will make you slightly happier
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    (Original post by SeanFM)
    Practice makes perfect

    And also confirming which ones will be in your formula booklet.
    That's true, practice does make perfect.

    (Original post by notnek)
    Check the Edexcel formula book (can Google it) and see which integrals are not there. These will be the ones you need to remember. Hopefully that will make you slightly happier
    Yeah I guess I will just have to remember the ones that are not in the formula book.
    Thank you, this seems less daunting now
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    (Original post by Philip-flop)
    That's true, practice does make perfect.


    Yeah I guess I will just have to remember the ones that are not in the formula book.
    Thank you, this seems less daunting now
    So it's all clear, you only need to know the first 5 of the integrals you posted and even those can be derived quickly in your head if you know the corresponding derivatives.

    You'll pick them up anyway as you do more questions. You're panicking too early in the chapter - C4 integration is hard but not yet!
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    (Original post by notnek)
    So it's all clear, you only need to know the first 5 of the integrals you posted and even those can be derived quickly in your head if you know the corresponding derivatives.

    You'll pick them up anyway as you do more questions.
    Yeah true. I'm not sure I remember exactly they are derived but at least there is less to remember than I initially though

    (Original post by notnek)
    You're panicking too early in the chapter - C4 integration is hard but not yet!
    I beg to differ! I'm already struggling with using the chain rule in reverse. Guess I just need to sit down and try to see things logically
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    (Original post by Philip-flop)
    Yeah true. I'm not sure I remember exactly they are derived but at least there is less to remember than I initially though


    I beg to differ! I'm already struggling with using the chain rule in reverse. Guess I just need to sit down and try to see things logically

    Some tips:

    Sit down and see if you prove the results for the main trigs listed yourself (look up the section in the textbook if you get stuck) as well as memorising them and repeat this exercise from time to time. It's easy to forget the role played by the double angle formula and basic trig identifies.

    The Solomon worksheets are great for improving your skills:
    http://www.physicsandmathstutor.com/...on-worksheets/

    Bear in mind that everyone has had to practice to get up to speed, that everyone apart from Tony Stark (who is fictional) makes mistakes, and that big improvments are possible!
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    (Original post by Philip-flop)
    So I've finally embarked onto the last chapter of C4 from the Edexcel Modular Textbook and already I am feeling quite overwhelmed! The reason being...
    Name:  C4 Ch.6 Integrating Standard Functions.png
Views: 213
Size:  74.6 KB

    Obviously I know I have technically covered all these as a lot of them are just the reverse process of what I learnt from the C3 Differentiation chapter. But even then I wasn't completely confident and feel like I will struggle to remember everything
    if you cant integrate you will disintegrate

    wise words to live by
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    (Original post by Philip-flop)
    Yeah true. I'm not sure I remember exactly they are derived but at least there is less to remember than I initially though


    I beg to differ! I'm already struggling with using the chain rule in reverse. Guess I just need to sit down and try to see things logically
    It's important to know that 'reverse chain rule' only works when the stuff in the brackets is linear. E.g. you can integrate \cos \left(3x+2\right) using reverse chain rule but not \cos \left(x^2 \right).

    This may be the number 1 most common mistake (aside from algebra) made by students in A Level maths. Some teachers don't teach 'reverse chain rule' at all for this reason - you can integrate instead using substitution, which you'll learn later on.
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    (Original post by BrasenoseAdm)
    Some tips:

    Sit down and see if you prove the results for the main trigs listed yourself (look up the section in the textbook if you get stuck) as well as memorising them and repeat this exercise from time to time. It's easy to forget the role played by the double angle formula and basic trig identifies.

    The Solomon worksheets are great for improving your skills:
    http://www.physicsandmathstutor.com/...on-worksheets/

    Bear in mind that everyone has had to practice to get up to speed, that everyone apart from Tony Stark (who is fictional) makes mistakes, and that big improvments are possible!
    Thank you!! Those worksheets look very helpful!

    (Original post by notnek)
    It's important to know that 'reverse chain rule' only works when the stuff in the brackets is linear. E.g. you can integrate \cos \left(3x+2\right) using reverse chain rule but not \cos \left(x^2 \right).

    This may be the number 1 most common mistake (aside from algebra) made by students in A Level maths. Some teachers don't teach 'reverse chain rule' at all for this reason - you can integrate instead using substitution, which you'll learn later on.
    Oh I see!! So it is still possible to integrate by using the method of substitution instead of the 'reverse chain rule'? How come the reverse chain rule is mentioned in the textbook if a lot of teachers don't end up teaching it anyway?
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    (Original post by Philip-flop)
    Thank you!! Those worksheets look very helpful!


    Oh I see!! So it is still possible to integrate by using the method of substitution instead of the 'reverse chain rule'? How come the reverse chain rule is mentioned in the textbook if a lot of teachers don't end up teaching it anyway?
    I think the majority of teachers do teach it. A lot of C4 integrals can be tackled using substitution but it's not the quickest method and it may not always be obvious what to substitute.

    Being able to "see" what an integral is immediately using reversal methods (you will see more useful methods later on in the chapter) without resorting to substitution will make your life easier and make you a more confident integrater!

    it's similar to how you should be able to see that the derivative of e^{2x} is 2e^{2x} or \cos 2x \rightarrow -2\sin 2x without going through the whole substitution and chain rule process.
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    (Original post by Philip-flop)
    So I've finally embarked onto the last chapter of C4 from the Edexcel Modular Textbook and already I am feeling quite overwhelmed! The reason being...
    Name:  C4 Ch.6 Integrating Standard Functions.png
Views: 213
Size:  74.6 KB

    Obviously I know I have technically covered all these as a lot of them are just the reverse process of what I learnt from the C3 Differentiation chapter. But even then I wasn't completely confident and feel like I will struggle to remember everything
    I can understand you. It is a quite tough lesson to remember. Integraion, the mathematical rules and solution steps to it. you have to practise a lot to get a routine in solving. There is no way around. But see it positive: when you have learnt it, it is not so difficult to get remember then. To speak for myself, it was more difficult to learn than to revise it. Just because of practising before. Be brave!
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    (Original post by notnek)
    I think the majority of teachers do teach it. A lot of C4 integrals can be tackled using substitution but it's not the quickest method and it may not always be obvious what to substitute.

    Being able to "see" what an integral is immediately using reversal methods (you will see more useful methods later on in the chapter) without resorting to substitution will make your life easier and make you a more confident integrater!

    it's similar to how you should be able to see that the derivative of e^{2x} is 2e^{2x} or \cos 2x \rightarrow -2\sin 2x without going through the whole substitution and chain rule process.
    Yeah I see what you mean. You just start being able do things without having to prove them the long way. I'll just keep knuckling down with this chapter for now then
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    This chapter is getting very difficult and I'm not even that far into it!


    The thing I've been struggling with the most is...

    Applying the general rule that  \int f' (ax+b) dx = \frac{1}{a} f(ax+b) in conjunction with trigonometric derivatives and using identities etc. For example when integrating something which looks as simple as...  \int sin^2 x dx I had trouble realising that I could rearrange the double angle formula  cos2x = 1 - 2sin^2 x \Rightarrow sin^2x = \frac{1}{2} - \frac{1}{2} cos 2x before being able to procede. But then I also had to remember the fact that  \frac{d(sinx}{dx} = cos x . These are all things that I really struggled with from C3 and am still trying to get to grips with, that's why I'm really struggling

    (Original post by Kallisto)
    I can understand you. It is a quite tough lesson to remember. Integraion, the mathematical rules and solution steps to it. you have to practise a lot to get a routine in solving. There is no way around. But see it positive: when you have learnt it, it is not so difficult to get remember then. To speak for myself, it was more difficult to learn than to revise it. Just because of practising before. Be brave!
    Thank you, I definitely need to carry on practising that's for sure! It's just really hard to get my head around it in the first place
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    (Original post by Philip-flop)
    (...)


    Thank you, I definitely need to carry on practising that's for sure! It's just really hard to get my head around it in the first place
    My advice to you: look at the function before you are trying to integrate. Then think about the integration rules and solution steps carefully. Especially the last one. If you have miscalculated by making mistakes in recognizing the term (that happened sometimes to me in integration by part!), all your doings were for nothing and that is very annoying for yourself. Especially you know that you can make it better. :sadnod:
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    (Original post by Philip-flop)
    This chapter is getting very difficult and I'm not even that far into it!


    The thing I've been struggling with the most is...

    Applying the general rule that  \int f' (ax+b) dx = \frac{1}{a} f(ax+b) in conjunction with trigonometric derivatives and using identities etc. For example when integrating something which looks as simple as...  \int sin^2 x dx I had trouble realising that I could rearrange the double angle formula  cos2x = 1 - 2sin^2 x \Rightarrow sin^2x = \frac{1}{2} - \frac{1}{2} cos 2x before being able to procede. But then I also had to remember the fact that  \frac{d(sinx}{dx} = cos x . These are all things that I really struggled with from C3 and am still trying to get to grips with, that's why I'm really struggling


    Thank you, I definitely need to carry on practising that's for sure! It's just really hard to get my head around it in the first place
    Firstly, it's important to realise that integration is not the same as differentiation where you can differentiate almost anything using the same set of methods. You will come across some expressions that seem simple but they may be very hard to integrate or it may not even be possible.

    In C4 you'll meet a variety of methods in order to tackle integrals but there are some exceptions which you'll need to get used to - there aren't many. \int \sin^2 x \ dx and \int \cos^2 x \ dx are exceptions.

    So far you have met the 'reverse chain rule' rule. That only works when the stuff in the brackets is linear. Looking at \sin^2 x, the first thing you could do is write it as \left(\sin x\right)^2. This cannot be integrated using reverse chain rule since the stuff in the brackets \sin x is not a linear function.

    So you need another method. But as I said before, there isn't a standard method that you can use; instead you need to use the trig identity and go from there as you've seen. You just need to practice this for the integrals of \sin^2 x and \cos^2 x.

    At this point you really need to be familiar with the derivatives and integrals of \sin x and \cos x. Set aside some time to learn these off-by-heart. It shouldn't take too long.
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    So I think my brain is fried after today as I can't seem to integrate this equation...

     \int \frac{1 - sin x}{cos^2 x}

    I notice that I can use the trig identity...  sin^2 x + cos^2 x = 1 \Rightarrow cos^2 x = 1 - sin^2 x to give...

     \int \frac{1 - sin x}{1 - sin^2 x} but then get stuck from there. I'm probably being stupid cos I'm tired but I can't seem to work this out

    (Original post by notnek)
    Firstly, it's important to realise that integration is not the same as differentiation where you can differentiate almost anything using the same set of methods. You will come across some expressions that seem simple but they may be very hard to integrate or it may not even be possible.

    In C4 you'll meet a variety of methods in order to tackle integrals but there are some exceptions which you'll need to get used to - there aren't many. \int \sin^2 x \ dx and \int \cos^2 x \ dx are exceptions.

    So far you have met the 'reverse chain rule' rule. That only works when the stuff in the brackets is linear. Looking at \sin^2 x, the first thing you could do is write it as \left(\sin x\right)^2. This cannot be integrated using reverse chain rule since the stuff in the brackets \sin x is not a linear function.

    So you need another method. But as I said before, there isn't a standard method that you can use; instead you need to use the trig identity and go from there as you've seen. You just need to practice this for the integrals of \sin^2 x and \cos^2 x.

    At this point you really need to be familiar with the derivatives and integrals of \sin x and \cos x. Set aside some time to learn these off-by-heart. It shouldn't take too long.
    Thanks notnek! I can always count on you for some reassurance and useful advice!
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    (Original post by Philip-flop)
    So I think my brain is fried after today as I can't seem to integrate this equation...

     \int \frac{1 - sin x}{cos^2 x}

    I notice that I can use the trig identity...  sin^2 x + cos^2 x = 1 \Rightarrow cos^2 x = 1 - sin^2 x to give...

     \int \frac{1 - sin x}{1 - sin^2 x} but then get stuck from there. I'm probably being stupid cos I'm tired but I can't seem to work this out


    Thanks notnek! I can always count on you for some reassurance and useful advice!
    Difference of two squares.
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    (Original post by NotNotBatman)
    Difference of two squares.
    OMG!! Why couldn't I see that?? Thank you so much!
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    (Original post by Philip-flop)
    OMG!! Why couldn't I see that?? Thank you so much!
    It might be easier though if you recall that \displaystyle \frac{a+b}{c}=\frac{a}{c} + \frac{b}{c}
 
 
 
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