How to do this integration?

[shaq]

Tw and σ^2 are constants.

 

[LouisSheffield]

The indefinite integral evaluates to
(π/2)^½ × σ × erf(x / (√2 × σ))

(sorry about these symbols)

Depending on x and Tw, you'll need either a good, or a better approximation to erf.

(i.e. (π/2)^½ × σ × (erf((x + Tw) / (√2 × σ)) - erf(x / (√2 × σ)))

Added after 11 minutes:

This might get you started on that last part ...
https://mathworld.wolfram.com/Erf.html

Search the web for approximations, and check your problem against values from tables to see if the accuracy's sufficient.

 

[shaq]

The indefinite integral evaluates to
(π/2)^½ × σ × erf(x / (√2 × σ))

(sorry about these symbols)

Depending on x and Tw, you'll need either a good, or a better approximation to erf.

(i.e. (π/2)^½ × σ × (erf((x + Tw) / (√2 × σ)) - erf(x / (√2 × σ)))

Added after 11 minutes:

This might get you started on that last part ...
https://mathworld.wolfram.com/Erf.html

Search the web for approximations, and check your problem against values from tables to see if the accuracy's sufficient.

Dear LouisSheffield,

What does the "n" mean?

 

[LouisSheffield]

It was trying to be a "pi"
Also, the square root applies only to the 2.

Added after 3 minutes:

I used MathCAD to do the integration.
Depending on definition, the wolfram site eludes to there being two competing definitions for erf(), depending on normalization.

When you get close (i.e. a routine for erf), give me a PM or an email, and we can figure out if your routine is normalized the same way as MathCAD's.

(I would hate for your answers to be off by a constant factor)

 

[belal saylan]

[color=darkred[b]](1/√(x^4+x^2+1)[[/[/b]color]quote="shaq"]Tw and σ^2 are constants.[/quote]∫

 

[dedede]

this integral could not be expressed by an elementary equation, could it?

 

[LouisSheffield]

Nope - only an approximation can, but the goodness of the approximation has to take many factors into account:
1) The value of x,
2) The value of Tw, and
3) The required precision of the approximation.

A form of this equation Q(x) is used often in calculating bit-error-rates. An approximation with an additive error of < 1% will hardly do for BERs on the order of 10^-16.