“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

If it’s an odd function y=y(x), then according to definition y(x)=-y(-x); and tailor series looking like y(x) = a1x +a2x^3 +a3x^5 + … exists, being all powers odd; after integration all powers become even, i.e. Y(x) = C +(a1/2)x^2 +(a2/4)x^4 + …;
Y(x)=Y(-x) is the feature of even function.

Answer 2

consider odd functions---
1) sinx. If u integrate it, u get -->-cosx..
2) x. If u integrate it, u get---->(x^2)/2, which is an even function..

So Integrating an odd function results in an even function

Answer 3

Read your other posts!!!!!!!!!!!!!!

Paste:

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 4

yes