Integral of (x-1)/ln(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 $du$:

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

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

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

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

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

            $$\int \frac{e^{\frac{2}{u}} - e^{\frac{1}{u}}}{u}\, du = - \int \frac{e^{\frac{2}{u}} - e^{\frac{1}{u}}}{u}\, du$$

            1. Rewrite the integrand:

               $$\frac{e^{\frac{2}{u}} - e^{\frac{1}{u}}}{u} = \frac{e^{\frac{2}{u}}}{u} - \frac{e^{\frac{1}{u}}}{u}$$

            2. Integrate term-by-term:

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

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

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

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

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

                     EiRule(a=2, b=0, context=exp(2*_u)/_u, symbol=_u)

                     So, the result is: $- \operatorname{Ei}{\left(2 u \right)}$

                  Now substitute $u$ back in:

                  $$- \operatorname{Ei}{\left(\frac{2}{u} \right)}$$

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

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

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

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

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

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

                        $$\int \frac{e^{u}}{u}\, du = - \int \frac{e^{u}}{u}\, du$$

                        EiRule(a=1, b=0, context=exp(_u)/_u, symbol=_u)

                        So, the result is: $- \operatorname{Ei}{\left(u \right)}$

                     Now substitute $u$ back in:

                     $$- \operatorname{Ei}{\left(\frac{1}{u} \right)}$$

                  So, the result is: $\operatorname{Ei}{\left(\frac{1}{u} \right)}$

               The result is: $\operatorname{Ei}{\left(\frac{1}{u} \right)} - \operatorname{Ei}{\left(\frac{2}{u} \right)}$

            So, the result is: $- \operatorname{Ei}{\left(\frac{1}{u} \right)} + \operatorname{Ei}{\left(\frac{2}{u} \right)}$

         Now substitute $u$ back in:

         $$\operatorname{Ei}{\left(2 u \right)} - \operatorname{Ei}{\left(u \right)}$$

      Now substitute $u$ back in:

      $$- \operatorname{Ei}{\left(\log{\left(x \right)} \right)} + \operatorname{Ei}{\left(2 \log{\left(x \right)} \right)}$$

   Method #2

   1. Rewrite the integrand:

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

   2. Integrate term-by-term:

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

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

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

         EiRule(a=2, b=0, context=exp(2*_u)/_u, symbol=_u)

         Now substitute $u$ back in:

         $$\operatorname{Ei}{\left(2 \log{\left(x \right)} \right)}$$

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

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

         LiRule(a=1, b=0, context=1/log(x), symbol=x)

         So, the result is: $- \operatorname{li}{\left(x \right)}$

      The result is: $\operatorname{Ei}{\left(2 \log{\left(x \right)} \right)} - \operatorname{li}{\left(x \right)}$

2. Add the constant of integration:

   $$- \operatorname{Ei}{\left(\log{\left(x \right)} \right)} + \operatorname{Ei}{\left(2 \log{\left(x \right)} \right)}+ \mathrm{constant}$$

The answer is:

$$- \operatorname{Ei}{\left(\log{\left(x \right)} \right)} + \operatorname{Ei}{\left(2 \log{\left(x \right)} \right)}+ \mathrm{constant}$$