The average of a function

[ribod]

I have this function:
y=x/(x-1/x)
and I want to find out the average value of y, between two values of x.

Is there some mathematical way to do this?

 

[cronxeh]

$$\frac{1}{x_{2}-x_{1}} \int_{x_{1}}^{x_{2}} {\frac{x}{x-\frac{1}{x}} dx }$$

$$a = x_{1}, b = x_{2}$$

$$\frac{1}{b-a} * ( b + \frac{1}{2}*log(b - 1) - \frac{1}{2}*log(b+1) - a - \frac{1}{2} * log(a-1) + \frac{1}{2} * log(a+1) )$$

 

[Zurtex]

Correct me if I am wrong but I think that nicely cancels down to:

$$\frac{1}{b - a}\left(b - a + \text{tanh}^{-1}(a) - \text{tanh}^{-1}(b) \right)$$

 

[cronxeh]

And you assume that if OP doesn't know the avg of a function, then he'll know what hyperbolic tangent is :rofl:

 

[DeadWolfe]

I know what tanh is but not the average of a function.

 

[Data]

Ah, but do you know what $$\tanh^{-1}$$ is??

 

[HallsofIvy]

The average of a function, f(x), between x= x₁ and x= x₂ is:
$$\frac{1}{x_2-x_1}\int_{x_1}^{x_2}f(x)dx$$