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# Definite Integration watch

1. 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.

2. (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.

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 for the integral from zero to of the periodic function .

Now consider and . Since has a period of , we have .

We can therefore write .

We can subtract from both sides and rearrange to get .

But from the first part, .

Thus will give you the constant in terms of the period .
3. 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
4. (Original post by Wilson Lui)

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.
5. Take for example.

You should be able to tell by considering the graph of that will have a period of .

We need to evaluate which you can do by first writing .

The term integrates to and will therefore make no contribution to the definite integral (since the limits of integration are 0 and ).

This leaves us with .

From before, so which is clearly a constant.

You can check for yourself that has a period of as required.

Note that I used as an example rather than the more familiar because for , you will get which is not very interesting.

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