Integral of 1*(lnx/2*lnx)*(1/x) dx

1. There are multiple ways to do this integral.

   Method #1

   1. Let $u = \log{\left(x \right)}$.

      Then let $du = \frac{dx}{x}$ and substitute $\frac{du}{2}$:

      $$\int \frac{u^{2}}{2}\, du$$

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

         $$\int u^{2}\, du = \frac{\int u^{2}\, du}{2}$$

         1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

            $$\int u^{2}\, du = \frac{u^{3}}{3}$$

         So, the result is: $\frac{u^{3}}{6}$

      Now substitute $u$ back in:

      $$\frac{\log{\left(x \right)}^{3}}{6}$$

   Method #2

   1. Let $u = \frac{1}{x}$.

      Then let $du = - \frac{dx}{x^{2}}$ and substitute $- \frac{du}{2}$:

      $$\int \left(- \frac{\log{\left(\frac{1}{u} \right)}^{2}}{2 u}\right)\, du$$

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

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

         1. Let $u = \frac{1}{u}$.

            Then let $du = - \frac{du}{u^{2}}$ and substitute $- du$:

            $$\int \left(- \frac{\log{\left(u \right)}^{2}}{u}\right)\, du$$

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

               $$\int \frac{\log{\left(u \right)}^{2}}{u}\, du = - \int \frac{\log{\left(u \right)}^{2}}{u}\, du$$

               1. Let $u = \log{\left(u \right)}$.

                  Then let $du = \frac{du}{u}$ and substitute $du$:

                  $$\int u^{2}\, du$$

                  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$:

                     $$\int u^{2}\, du = \frac{u^{3}}{3}$$

                  Now substitute $u$ back in:

                  $$\frac{\log{\left(u \right)}^{3}}{3}$$

               So, the result is: $- \frac{\log{\left(u \right)}^{3}}{3}$

            Now substitute $u$ back in:

            $$\frac{\log{\left(u \right)}^{3}}{3}$$

         So, the result is: $- \frac{\log{\left(u \right)}^{3}}{6}$

      Now substitute $u$ back in:

      $$\frac{\log{\left(x \right)}^{3}}{6}$$

2. Add the constant of integration:

   $$\frac{\log{\left(x \right)}^{3}}{6}+ \mathrm{constant}$$

The answer is:

$$\frac{\log{\left(x \right)}^{3}}{6}+ \mathrm{constant}$$