# Anti-derivatives of the periodic functions

• MHB
• cbarker1
In summary, an anti-derivative of a periodic function is a function whose derivative is equal to the original periodic function, plus a constant. To find the anti-derivative, one can use basic rules of integration and trigonometric identities. However, not all periodic functions have anti-derivatives, such as functions without a well-defined derivative. In real-world applications, anti-derivatives are useful for understanding and predicting the behavior of phenomena modeled by periodic functions. There is no general formula for finding the anti-derivative, but there are common techniques and identities that can be used.
cbarker1
Gold Member
MHB
Dear Everyone,

I do not know how to begin with the following problem:Suppose that $f$ is $2\pi$-periodic and let $a$ be a fixed real number. Define $F(x)=\int_{a}^{x} f(t)dt$, for all $x$ .
Show that $F$ is $2\pi$-periodic if and only if $\int_{0}^{2\pi}f(t)dt=0$.
Thanks,
Cbarker1

Last edited:
Cbarker1 said:
...if and only if $\int_{0}^{2\pi}f(t)dt$.
if and only if $$\displaystyle \int_0^{2 \pi}f(t)~dt$$ is what?

-Dan

## 1. What are anti-derivatives of periodic functions?

Anti-derivatives of periodic functions are functions that, when differentiated, result in the original periodic function. They are also known as indefinite integrals.

## 2. How do you find the anti-derivative of a periodic function?

To find the anti-derivative of a periodic function, you can use the fundamental theorem of calculus, which states that the anti-derivative of a function can be found by integrating the function with respect to the variable and adding a constant of integration.

## 3. Can all periodic functions have anti-derivatives?

Yes, all periodic functions have anti-derivatives. This is because the derivative of a periodic function is always a continuous function, and all continuous functions have anti-derivatives.

## 4. Are anti-derivatives of periodic functions unique?

No, anti-derivatives of periodic functions are not unique. This is because adding a constant of integration to the anti-derivative does not change its derivative, so there can be infinite anti-derivatives of a periodic function.

## 5. How are anti-derivatives of periodic functions used in real-life applications?

Anti-derivatives of periodic functions are used in various fields of science and engineering, such as physics, chemistry, and electrical engineering. They are used to model and analyze periodic phenomena, such as the motion of objects, electrical signals, and chemical reactions.

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