The following link shows the indefinite integral: http://www.wolframalpha.com/input/?i=int+ln(2-2*cos(x))+dx The following two links generate the definite integrals (at first it just shows the numeric value, but it you give it a minute to update, it shows the symbolic form): http://www.wolframalpha.com/input/?i=int+ln(2-2*cos(x))+dx,+x%3D0..pi%2F3 http://www.wolframalpha.com/input/?i=int+ln(2-2*cos(x))+dx,+x%3Dpi%2F3..pi Tom Tom Karzes writes:
I tried this (for free) on Wolfram Alpha, and it says the same thing. It gives the following expressions for the two integrals:
-1/18 i (π^2 - 36 Li_2((-1)^(1/3))) 1/18 i (π^2 - 36 Li_2((-1)^(1/3)))
which clearly sum to zero. (It says Li_n is the polylogarithm function.)
Tom
Dan Asimov writes:
Thanks! (That HAD to be true since no two real numbers can be as close as these were.)
—Dan
According to maple they are symbolically equal. I did not dig into it to try to see why.
On Mon, Aug 20, 2018 at 5:19 PM Dan Asimov <dasimov@earthlink.net> wrote:
Let f(x) = ln(2-2*cos(x))
Then are these two numbers equal?
A = -Integral from 0 to π/3 of f(t) dt
B = Integral from π/3 to π of f(t) dt
I can't tell. But they're extremely close.
—Dan