Question

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

Answer 1

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.