Integrating an odd function results in an even function”. Is it true?Justify.?

Question

(Note: When you integrate sin x you get -cos x which is even)

Answer 1

One way is Taylor's series.

If it is an analytic function (Taylor series convereges to f(x)) then an odd function has only odd degree terms and so when you integerate term by term you get even degree terms which is then an even function.

Otherwise, let f(t) be an odd function, and let F(x)=int(f(t),t=0 to x,dt) be an antiderivative. Then F(-x) = int(f(t), t=0..-x, dt) = int(f(-t), t=-0..-(-x), d(-t)) = int(-f(t), t=0..x, -dt) = int(f(t), t=0..x, dt) = F(x). So basically you get a negative sign from the dt part and a negative sign from the odd function part and those cancel.

Answer 2

every time it isn`t correct, but some it is correct
eg:sinx cosx eg: ex ex
cosx -sinx
tanx secx
secx secx tanx

Answer 3

Yes, it is true.

By definition,
f'(x) = lim[f(y)-f(x)]/[y-x]

If f(x) is odd, then
f'(-x) = lim[f(-y)-f(-x)]/[-y+x] = lim[f(y)-f(x)]/[y-x] = f'(x)

Can you finish it now?