Find The Indefinite Integral. (the Anti-derivative)?

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1) int ln(sin(x)) dx
Please see:
http://www.forkosh.dreamhost.com/mimetex.cgi?\int%20\ln(\sin(x))%20dx
2) int ln(x)/(x+k)dx (k in lR)
Please see:
http://www.forkosh.dreamhost.com/mimetex.cgi?\int\frac{\ln(x)}{x+k}dx\qquad%20k\in%20\mathbb{R}

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"Find The Indefinite Integral. (the Anti-derivative)?" was posted on Wednesday, July 8th, 2009 at 12:37 am.

2 Responses to “Find The Indefinite Integral. (the Anti-derivative)?”

Dr D(2009-07-08):

They have no closed form analytical anti derivative.
However definite integrals can be computed using Fourier transforms, complex variables or what have you.
1,1,2,3,3,4, 5,5,6,6,6, 8,8,8,10(2009-07-08):

-x*ln(1-exp((2*i)*x)) + x*ln(sin(x)) + (1/2)*i*x^2 +
(1/2)*i*polylog(2, exp((2*i)*x))
dilog((x+k)/k) + ln(x)*ln((x+k)/k)
Polylog and Dilog are infinite sums.