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    On the last chapter of C2, as I finish this self-taught module, and wondering what is the result of integrating f(x)? How does that work? I thought the purpose of integration was to reverse from the f'(x) function into f(x). But in Chapter 11, f(x) functions are being integrated, and whilst I can do the exercises, I don't quite understand what I'm actually doing... if anyone could explain what integrating f(x) gives you... in comprehendable english...
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    You're thinking about it too much. You're just integrating another function. If you really want, you can call it f'(x).
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    At C2 all you need to know is that, in some sense, "integration is the reverse of differentiation". When you integrate f(x), you get a function g(x), say, such that g'(x)=f(x)... so if it helps, write f(x)=g'(x) and then find g(x).

    For example, if you have to integrate x^n you need to think "what differentiates to give this?", and instinct should point to some multiple of x^{n+1}. Differentiating this gives (n+1)x^n, which is n+1 times too much, and so we need \dfrac{x^{n+1}}{n+1} instead (which you can check does differentiate to give the right thing). But because constants differentiate to zero, in order to be completely general we say that \displaystyle \int x^n\, dx = \dfrac{x^{n+1}}{n+1}+C where C is some arbitrary constant.

    If you want more information than this then you'll need to give an example. Many functions can't even be integrated, and even many functions that can be integrated can't be expressed in terms of elementary functions (by which I mean things like powers, exponentials, logarithms, trig functions, hyperbolic functions). [EDIT: But in C2 everything can be integrated and expressed in terms of elementary functions, so don't worry about this. Like I say, examples of questions you're stuck on would be useful for us to help you.]
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    (Original post by nuodai)
    At C2 all you need to know is that, in some sense, "integration is the reverse of differentiation". When you integrate f(x), you get a function g(x), say, such that g'(x)=f(x)... so if it helps, write f(x)=g'(x) and then find g(x).

    For example, if you have to integrate x^n you need to think "what differentiates to give this?", and instinct should point to some multiple of x^{n+1}. Differentiating this gives (n+1)x^n, which is n+1 times too much, and so we need \dfrac{x^{n+1}}{n+1} instead (which you can check does differentiate to give the right thing). But because constants differentiate to zero, in order to be completely general we say that \displaystyle \int x^n\, dx = \dfrac{x^{n+1}}{n+1}+C where C is some arbitrary constant.

    If you want more information than this then you'll need to give an example. Many functions can't even be integrated, and even many functions that can be integrated can't be expressed in terms of elementary functions (by which I mean things like powers, exponentials, logarithms, trig functions, hyperbolic functions). [EDIT: But in C2 everything can be integrated and expressed in terms of elementary functions, so don't worry about this. Like I say, examples of questions you're stuck on would be useful for us to help you.]
    Thank you for your attempt to explain it.

    I'm not really stuck on anything, I'm just curious about what the result of integrating f(x) gives you.

    For example, integrating f'(x) gives you f(x), i.e. a transformation from f'(x) to f(x).
    So my real query is if we integrate f(x), what do we get? Just f(x) again?
    - integral of f(x)=9x^2+4x gives you 3x^3+2x^2 <------ this result, what is the notation for it? f(x)=3x^3 etc or something else.
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    (Original post by snow leopard)
    Thank you for your attempt to explain it.

    I'm not really stuck on anything, I'm just curious about what the result of integrating f(x) gives you.

    For example, integrating f'(x) gives you f(x), i.e. a transformation from f'(x) to f(x).
    So my real query is if we integrate f(x), what do we get? Just f(x) again?
    - integral of f(x)=9x^2+4x gives you 3x^3+2x^2 <------ this result, what is the notation for it? f(x)=3x^3 etc or something else.
    Ah right... well the notation for it is simply \displaystyle \int f(x)\, dx. Although there's the shorthand notation f'(x) for \dfrac{d}{dx}f(x), we don't have this luxury for integration.

    There's a reason for this. When you have a function, it always has just one derivative; for example if f(x)=x^5+2x then f'(x)=5x^4+2. It doesn't differentiate to anything else.

    However, functions have more than one integral. For example, because both x^3 and x^3 + 1 differentiate to give the same thing, 3x^2, the integral of 3x^2 could be x^3 or x^3+1 or indeed x^3+\text{anything} (as long as the 'anything' is constant).

    This is important because if you integrate 3x^2 twice then we can get anything in the form \dfrac{x^4}{4} + ax + b where a and b are any constants, and so our functions no longer differ just by a constant, they differ by a linear polynomial... and so on.
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    (Original post by nuodai)
    Ah right... well the notation for it is simply \displaystyle \int f(x)\, dx. Although there's the shorthand notation f'(x) for \dfrac{d}{dx}f(x), we don't have this luxury for integration.

    There's a reason for this. When you have a function, it always has just one derivative; for example if f(x)=x^5+2x then f'(x)=5x^4+2. It doesn't differentiate to anything else.

    However, functions have more than one integral. For example, because both x^3 and x^3 + 1 differentiate to give the same thing, 3x^2, the integral of 3x^2 could be x^3 or x^3+1 or indeed x^3+\text{anything} (as long as the 'anything' is constant).

    This is important because if you integrate 3x^2 twice then we can get anything in the form \dfrac{x^4}{4} + ax + b where a and b are any constants, and so our functions no longer differ just by a constant, they differ by a linear polynomial... and so on.
    There we go :flip: muchas gracias ~
 
 
 
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