I was solving a problem relating to anti derivative and integrals which is as follows:
Ex. Suppose f(x) is an periodic function with period equal to p>0 and it is integrable on interval [0,p]. prove:
$$\int_{a}^{p+a}f(x)dx=\int_{0}^pf(x)dx$$
and while solving I found a way much shorter than the one the book provided and my solution looks like this:
according to the fundamental theorem of calculus there exists a function like G that when derivated will be equal to f(x) so we have:
$$\int_{a}^{p+a}f(x)=G(p+a)-G(a)$$
now if G is also periodic we can say G(p+a)=G(a) therefor the mentioned integral is equal to zero then we calculate the second integral:
$$\int_{0}^{p}f(x)=G(p)-G(0)$$
now again if G is periodic and has the same period as f the second integral is also zero and equal to the first integral so the question is answered.
but my solution requires an answer to two questions:
Can we prove that the antiderivative of an periodic function also periodic?
if so are the periods of two functions the same?
thanks in advance.
Best Answer
[Comment elevated to answer, at request of OP.]
The antiderivative of a periodic function need not be periodic. For example, $f(x)=1+\sin x$ is periodic, but the antiderivative $x-\cos x$ is not.