Integral of x*log3(x) dx

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

   Then let $du = \frac{dx}{x}$ and substitute $\frac{du}{\log{\left(3 \right)}}$:

   $$\int \frac{u e^{2 u}}{\log{\left(3 \right)}}\, du$$

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

      $$\int u e^{2 u}\, du = \frac{\int u e^{2 u}\, du}{\log{\left(3 \right)}}$$

      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 $\operatorname{dv}{\left(u \right)} = e^{2 u}$.

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

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

         1. Let $u = 2 u$.

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

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

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

               $$\text{False}$$

               1. The integral of the exponential function is itself.

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

               So, the result is: $\frac{e^{u}}{2}$

            Now substitute $u$ back in:

            $$\frac{e^{2 u}}{2}$$

         Now evaluate the sub-integral.

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

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

         1. Let $u = 2 u$.

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

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

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

               $$\text{False}$$

               1. The integral of the exponential function is itself.

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

               So, the result is: $\frac{e^{u}}{2}$

            Now substitute $u$ back in:

            $$\frac{e^{2 u}}{2}$$

         So, the result is: $\frac{e^{2 u}}{4}$

      So, the result is: $\frac{\frac{u e^{2 u}}{2} - \frac{e^{2 u}}{4}}{\log{\left(3 \right)}}$

   Now substitute $u$ back in:

   $$\frac{\frac{x^{2} \log{\left(x \right)}}{2} - \frac{x^{2}}{4}}{\log{\left(3 \right)}}$$

2. Now simplify:

   $$\frac{x^{2} \left(2 \log{\left(x \right)} - 1\right)}{4 \log{\left(3 \right)}}$$

3. Add the constant of integration:

   $$\frac{x^{2} \left(2 \log{\left(x \right)} - 1\right)}{4 \log{\left(3 \right)}}+ \mathrm{constant}$$

The answer is:

$$\frac{x^{2} \left(2 \log{\left(x \right)} - 1\right)}{4 \log{\left(3 \right)}}+ \mathrm{constant}$$