What is the integration of #1/log(sqrt(1-x))# ?

1 Answer
A. S. Adikesavan
May 27, 2016

Here, log is ln.. Answer:#(2sum((-1)^(n-1))/n(x/ln(1-x))^n, n =1, 2, 3, ..oo)# + C..
#=2ln(1+x/(ln(1-x)))+C, |x/(ln(1-x))|<1#

Explanation:

Use #intu dv = uv-intv du#, successively.

#inti/(lnsqrt(1-x) dx#

#=2int1/ln(1-x) dx#

#=2[x/ln(1-x)-intxd(1/ln(1-x))]#

#=2[[x/ln(1-x)-intx/(ln(1-x))^2 dx]#

#=2[[x/ln(1-x)-int1/(ln(1-x))^2 d(x^2/2)]#

and so on.

The ultimate infinite series appears as answer.

I am yet to study the interval of convergence for the series.

As of now, # |x/(ln(1-x) )|<1#

The explicit interval for x, from this inequality, regulates the interval for any definite integral for this integrand. Perhaps, I might give this, in my 4th edition of the answer.

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