Pfft, you call yourself a programmer, but don't know how to solve the formula one?
Everybody knows programmers don't have to give the *right* answer, they just have to give something good enough to satisfy the needs of the clients. Someone who wants you to find that number probably needs the value for a particular practical purpose, not "just for kicks"
So, the following Java snippet:
Double x = 1.0;
Double a = 0.0;
while (x > 0.0000000000000000000000000000001)
{
a = Math.log(2 + Math.sqrt((Math.atan(x) * Math.sin((1/x)))));
System.out.println(a);
x = x / 1.1;
}
Allows you approximate A (the needed answer), as X approaches 0.
Yielding the value 0.6931471805599453
Since A is an irrational and transcendental number, you cannot express it exactly anyways, so the only way to simplify it to a decimal is to give an approximation with arbitrary precision -- here limited by the size of the Java Double, which is probably precise enough for any real-life purpose.