# Prove F(-x) = -F(x) for all x

1. Oct 21, 2011

### Noorac

f(x)
=$\frac{arctan x}{x}$ for x different from 0
= 1 for x equal to 0

F(x) is a definite integral from 0 to x, but I couldn't find the code for it, so just assume it is from 0 to x in the equation below.

F(X) = $\int f(t) dt$

Now, the task is to prove that F(-x) = -F(x).

This means we need to prove:

$\int f(t) dt$ : definite from 0 to -x

=

-$\int f(t) dt$ : definite from 0 to x
_____________________________________

We have tried some basic manipulation of integrals, but came nowhere.

If anyone can give a hint of how to prove this, or tell us if we have understood the problem wrong, we would be most grateful.

Thanks.

2. Oct 21, 2011

### SammyS

Staff Emeritus
The way to include limits of integration in LaTeX code is to use the subscript & superscript with the integral symbol: \int_{0}^{x} in your case. Using the \displaystyle code will make Large integral symbols and fractions when using [​itex​] & [​/itex​] tags.

The code:
Code (Text):
$\displaystyle -\int_{0}^{x} f(t) dt$
gives you $\displaystyle -\int_{0}^{x} f(t) dt$ .

3. Oct 21, 2011

### dynamicsolo

A quick way to look at this: what is the symmetry (about the y-axis) of the function $\frac{arctan x}{x}$? (It is the ratio of two functions -- what are their symmetries and so what is the symmetry of the ratio?). Then, what does it mean to integrate $\int_{0}^{x} f(t) dt$ and $\int_{0}^{-x} f(t) dt$ for a function f(x) with such a symmetry?