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    There is a question about my integration assignment.
    Let f(x) be a periodic continuous function with a period of T and a be a real number.

    (a) [proved already] prove that the integral from a to (a+T) of f(x)dx = the integral from 0 to T of f(x)dx

    (b) HENCE find a constant C such that the function g(x) = Cx + the integral from 0 to x of f(t)dt is a periodic function with a period of T.

    Please help!
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    (Original post by Wilson Lui)
    There is a question about my integration assignment.
    Let f(x) be a periodic continuous function with a period of T and a be a real number.

    (a) [proved already] prove that the integral from a to (a+T) of f(x)dx = the integral from 0 to T of f(x)dx

    (b) HENCE find a constant C such that the function g(x) = Cx + the integral from 0 to x of f(t)dt is a periodic function with a period of T.

    Please help!
    This is a very interesting question. I am going to give it a go, but forgive me if I am mistaken.

    I will use the notation F(x) for the integral from zero to x of the periodic function f.

    Now consider g(a) and g(a + T). Since g has a period of T, we have g(a) = g(a + T).

    We can therefore write aC + F(a) = aC + TC + F(a + T).

    We can subtract aC from both sides and rearrange to get TC + F(a + T) - F(a) = 0.

    But from the first part, F(a + T) - F(a) = F(T).

    Thus TC + F(T) = 0 will give you the constant C in terms of the period T.
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    Thanks for your reply. Indeed I've gone to this step too.

    But then I thought of the requirements: Find a CONSTANT C such that...

    I wonder if it is appropriate if the constant is still in terms of other variables such as T.

    Hence I wonder if there are also other methods. ...

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    (Original post by Wilson Lui)
    Thanks for your reply. Indeed I've gone to this step too.

    But then I thought of the requirements: Find a CONSTANT C such that...

    I wonder if it is appropriate if the constant is still in terms of other variables such as T.

    Hence I wonder if there are also other methods. ...

    Posted from TSR Mobile
    Well once you choose your periodic function, its period won't be variable. You will get some fixed value for T for that function, which in turn will define a fixed value for C.
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    Take f(x) = sin^2(x) for example.

    You should be able to tell by considering the graph of y = sin(x) that f will have a period of \pi.

    We need to evaluate F(\pi) which you can do by first writing sin^2(x) = \frac{1}{2} (1 - cos(2x)).

    The cos(2x) term integrates to \frac{1}{2} sin(2x) and will therefore make no contribution to the definite integral (since the limits of integration are 0 and \pi).

    This leaves us with F(\pi) = \frac{\pi}{2}.

    From before, \pi C + F(\pi) = 0 so C = - \frac{1}{2} which is clearly a constant.

    You can check for yourself that g(x) = - \frac{x}{2} + F(x) has a period of \pi as required.

    Note that I used sin^2(x) as an example rather than the more familiar sin(x) because for f(x) = sin(x), you will get C = 0 which is not very interesting.
 
 
 
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