Integral of (3ln^3x)/(x)dx 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 $3 du$:

      $$\int 3 u^{3}\, du$$

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

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

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

            $$\int u^{3}\, du = \frac{u^{4}}{4}$$

         So, the result is: $\frac{3 u^{4}}{4}$

      Now substitute $u$ back in:

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

   Method #2

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

      Then let $du = - \frac{dx}{x^{2}}$ and substitute $- 3 du$:

      $$\int \left(- \frac{3 \log{\left(\frac{1}{u} \right)}^{3}}{u}\right)\, du$$

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

         $$\int \frac{\log{\left(\frac{1}{u} \right)}^{3}}{u}\, du = - 3 \int \frac{\log{\left(\frac{1}{u} \right)}^{3}}{u}\, du$$

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

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

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

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

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

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

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

                  $$\int u^{3}\, du$$

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

                     $$\int u^{3}\, du = \frac{u^{4}}{4}$$

                  Now substitute $u$ back in:

                  $$\frac{\log{\left(u \right)}^{4}}{4}$$

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

            Now substitute $u$ back in:

            $$- \frac{\log{\left(u \right)}^{4}}{4}$$

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

      Now substitute $u$ back in:

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

2. Add the constant of integration:

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

The answer is:

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