(In reply to numerical integration with computer--but what is this number?
Using Wolfram's Online Integrator, you have to use x rather than y and ln is just log, but, converted to use of y and ln, the antiderivative given there is:
-ln(sin(y)) * cos(y) + cos(y) + ln(tan(y/2))
Evaluated at y = pi/2, this comes out to just zero.
But evaluated at y = 0, the first of the three terms would become positive infinity and the last term negative infinity, making the middle term of 1 useless.
One would have to seek the limit of this as y approaches zero.
cos(y) approaches 1 obviously.
ln(tan(y/2)) - ln(sin(y)) = ln(tan(y/2)/sin(y))
Since tan(y)/sin(y) approaches 1, tan(y/2)/sin(y) approaches 1/2.
So the definite integral would be 0 - (1 + ln(1/2)) = -1 - ln(1/2) = ln(2) - 1
ln(2) ~= .6931471805599453
so the integral does evaluate to -0.3068528194400547
Posted by Charlie
on 2009-05-19 13:09:31