Integral of e^xlnx 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 $du$:

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

      1. Use integration by parts:

         $$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$

         Let $u{\left(u \right)} = u$ and let $\operatorname{dv}{\left(u \right)} = e^{u} e^{e^{u}}$.

         Then $\operatorname{du}{\left(u \right)} = 1$.

         To find $v{\left(u \right)}$:

         1. Let $u = e^{u}$.

            Then let $du = e^{u} du$ and substitute $du$:

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

            1. The integral of the exponential function is itself.

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

            Now substitute $u$ back in:

            $$e^{e^{u}}$$

         Now evaluate the sub-integral.

      2. Let $u = e^{u}$.

         Then let $du = e^{u} du$ and substitute $du$:

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

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

         Now substitute $u$ back in:

         $$\operatorname{Ei}{\left(e^{u} \right)}$$

      Now substitute $u$ back in:

      $$e^{x} \log{\left(x \right)} - \operatorname{Ei}{\left(x \right)}$$

   Method #2

   1. Use integration by parts:

      $$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$

      Let $u{\left(x \right)} = \log{\left(x \right)}$ and let $\operatorname{dv}{\left(x \right)} = e^{x}$.

      Then $\operatorname{du}{\left(x \right)} = \frac{1}{x}$.

      To find $v{\left(x \right)}$:

      1. The integral of the exponential function is itself.

         $$\int e^{x}\, dx = e^{x}$$

      Now evaluate the sub-integral.

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

2. Add the constant of integration:

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

The answer is:

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