Question

Define F(x)=integral from a to x of f(t)dt for all x. show that F is 2pi-periodic function if and only if the integral from 0 to 2pi of f(t)dt=0

Answer 1

\[F(x)=\int\limits_{a}^{x}f(t)dt for all x. Show that F is 2\Pi periodic if and only if \int\limits_{0}^{2\Pi}dt=0\]

Answer 2

For all \(x\), let \(F(x)=\displaystyle\int\limits_{a}^{x}f(t) \,dt \) .
Show that \( F\) is 2\(\pi\) periodic if and only if \(\displaystyle\int\limits_{0}^{2\pi} f(t) \,dt=0\).

Answer 3

Suppose \(F\) is \(2\pi\) periodic. Then \(F(0) = F(2\pi)\), So \(F(0) - F(2\pi) = 0\). Now use Fund. Thm of Calculus to show that the integral is 0.

Answer 4

Then assume that the integral is 0, and use the FTC again.

Answer 5

???

Answer 6

thank you!!! got it :)