The discussion revolves around a problem involving the integration of the natural logarithm of the sine function, specifically ln(sin(x)). Participants are exploring the challenges associated with finding the primitive (antiderivative) of sin(x) and the implications of the integral's limits.
Some participants have provided hints and clues regarding the integral, while others express confusion about the symmetry of the functions involved. The discussion is ongoing, with various interpretations and approaches being explored without a clear consensus.
There is mention of the integral not having an elementary solution and the need to consider the behavior of the logarithm function within specific limits. Participants also note the importance of understanding the properties of the sine function and its derivative.
Dick said:This is basically a trick. There is no elementary primitive. Here's a clue. Change the range of integration to 0 to pi/2 and call the integral I. Then you want -2*I. Now observe the integral of log(cos(x)) is also I. That's the clue. Add integral log(sin(x)) and log(cos(x)) and use a rule of logarithms and a trig identity and a u-substitution. Now you got an equation with a bunch of I's in it. Can you solve for I?
Dick said:They are very different. But they are still symmetric around x=pi/2. Did you solve the problem? It's really not that hard if you put your mind to it and know the secret hint. I only knew it because I've seen this problem before.
heyman123 said:i can do it but i need the primitive of sin x and that's my problem, the rest of the problem i can solve , i just only can't manage to discover de primitive of sin x
HallsofIvy said:Surely, you didn't get to where you are being expect to solve problems like this without learning that the derivative of cos x is -sin x??