Integral of (ln2)/((1-x)*(ln(1-x))^2) dx

1. The integral of a constant times a function is the constant times the integral of the function:

   $$\int \frac{\log{\left(2 \right)}}{\left(1 - x\right) \log{\left(1 - x \right)}^{2}}\, dx = \log{\left(2 \right)} \int \frac{1}{\left(1 - x\right) \log{\left(1 - x \right)}^{2}}\, dx$$

   1. There are multiple ways to do this integral.

      Method #1

      1. Rewrite the integrand:

         $$\frac{1}{\left(1 - x\right) \log{\left(1 - x \right)}^{2}} = - \frac{1}{x \log{\left(1 - x \right)}^{2} - \log{\left(1 - x \right)}^{2}}$$

      2. The integral of a constant times a function is the constant times the integral of the function:

         $$\int \left(- \frac{1}{x \log{\left(1 - x \right)}^{2} - \log{\left(1 - x \right)}^{2}}\right)\, dx = - \int \frac{1}{x \log{\left(1 - x \right)}^{2} - \log{\left(1 - x \right)}^{2}}\, dx$$

         1. Don't know the steps in finding this integral.

            But the integral is

            $$- \frac{1}{\log{\left(1 - x \right)}}$$

         So, the result is: $\frac{1}{\log{\left(1 - x \right)}}$

      Method #2

      1. Rewrite the integrand:

         $$\frac{1}{\left(1 - x\right) \log{\left(1 - x \right)}^{2}} = \frac{1}{- x \log{\left(1 - x \right)}^{2} + \log{\left(1 - x \right)}^{2}}$$

      2. Rewrite the integrand:

         $$\frac{1}{- x \log{\left(1 - x \right)}^{2} + \log{\left(1 - x \right)}^{2}} = - \frac{1}{x \log{\left(1 - x \right)}^{2} - \log{\left(1 - x \right)}^{2}}$$

      3. The integral of a constant times a function is the constant times the integral of the function:

         $$\int \left(- \frac{1}{x \log{\left(1 - x \right)}^{2} - \log{\left(1 - x \right)}^{2}}\right)\, dx = - \int \frac{1}{x \log{\left(1 - x \right)}^{2} - \log{\left(1 - x \right)}^{2}}\, dx$$

         1. Don't know the steps in finding this integral.

            But the integral is

            $$- \frac{1}{\log{\left(1 - x \right)}}$$

         So, the result is: $\frac{1}{\log{\left(1 - x \right)}}$

   So, the result is: $\frac{\log{\left(2 \right)}}{\log{\left(1 - x \right)}}$

2. Add the constant of integration:

   $$\frac{\log{\left(2 \right)}}{\log{\left(1 - x \right)}}+ \mathrm{constant}$$

The answer is:

$$\frac{\log{\left(2 \right)}}{\log{\left(1 - x \right)}}+ \mathrm{constant}$$