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

[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.

 

[SammyS]

f(x)
= $\displaystyle \frac{\arctan(x)}{x}$ 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) = $\displaystyle \int_{0}^{x} f(t) dt$

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

This means we need to prove:

$\displaystyle \int_{0}^{-x} f(t) dt$ : definite from 0 to -x

=

$\displaystyle -\int_{0}^{x} f(t) dt$ : 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.

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:

[itex]\displaystyle -\int_{0}^{x} f(t) dt[/itex]

gives you $\displaystyle -\int_{0}^{x} f(t) dt$ .

 

[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?