Question

integration of ln(sinx) ?

Answer 1

i can help if its an definite integral from 0 to pi/2

Answer 2

or from 0 to pi

Answer 3

indefinite integral of ln sinx cannot be expressed in terms of standard function

Answer 4

@hartnn and @laibashah : Have you considered integration by parts?

Supposing that we chose u=lin(sin x) and dv=dx, v=x and \[du=\frac{ 1 }{ \sin x }\cos x dx=\cot x dx\],

Answer 5

and so we'd obtain (ln sin x)(x) - Int(x*cot x dx), which itself could, I believe, be integrated through integration by parts.

Answer 6

integral x cot x is again, not integrable in terms of standard functions

Answer 7

i proposed a much easier approach here, (easier than multiple by parts integration), but only if the limits are from 0 to pi or 0 to pi/2

http://openstudy.com/users/aliza_k#/updates/5039a6e2e4b043c156a31ff9

Answer 8

It did dawn upon me that integrating cot x would result in an expression involving the ln function again, which seems to support your contention that x cot x is not integrable in terms of standard functions, unless you'd consider converting cot x to a Taylor series, multiplying that Taylor series by x, and then integrating term by term. Would you consider that approach to be "in terms of standard functions" or not?

Answer 9

then you'll get an infinite series again,
which will not have standard function representation.

Answer 10

Yes, that makes sense. Thanks for the intelligent discussion. By the way, do you have a favorite definition of "not integrable in terms of standard functions" that would be comprehensible to a relative lay person?

Answer 11

definition ?

standard functions : Trigonometric(sin/cos/tan....) , Exponential/Log function, Hyperbolic Functions, Gamma/Beta functions, Algebraic functions.....and all the functions taught in colleges :P

not integrable in terms of standard functions = answer is not among those standard functions

Answer 12

there is no explicit answer so i can solve it iat the interval [0 , pi/2]