Find indefinite integral function, if definite integral value is know

[jam_27]

Is this possible..

Say, _a∫^b f(x)dx = G(x)|_x=b - G(x)|_x=a = S, where S, a and b are known. Can we find G(x) ?

 

[SteamKing]

Not always. There are many cases where f(x) is known quite well, but the indefinite integral G(x) cannot be expressed using primitive functions or combinations of primitive functions. The definite integrals of such functions can nevertheless be computed to high-precision. A commonly encountered example of such a function describes the normal probability distribution:

http://en.wikipedia.org/wiki/Normal_distribution

The cdf for the normal probability distribution is given in terms of a special function called the error function, or erf (x) for short. erf (x) cannot be expressed in terms of logs, exponentials, etc., but tables of its values can be computed and used. There are many other examples of such functions, like elliptic functions or Bessel functions, etc.

 

[jam_27]

Ok, then, how much can we proceed to? I mean can we write G (x) in terms of S, a and b plus some unknown?
Also, b=1 and a= 0 and S is a constant in my case.

 

[SteamKing]

Ok, then, how much can we proceed to? I mean can we write G (x) in terms of S, a and b plus some unknown?
Also, b=1 and a= 0 and S is a constant in my case.

All you know for certain is that G(1) - G(0) = S. There may be many different functions f(x) the definite integral of which between x = b and x = a will give the same result. Unless you have some further conditions or restrictions, I don't think you can determine f(x) uniquely.