Integral of log^3(x) dx

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

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

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

   1. Use integration by parts:

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

      Let $u{\left(u \right)} = u^{3}$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$.

      Then $\operatorname{du}{\left(u \right)} = 3 u^{2}$.

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

      1. The integral of the exponential function is itself.

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

      Now evaluate the sub-integral.

   2. Use integration by parts:

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

      Let $u{\left(u \right)} = 3 u^{2}$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$.

      Then $\operatorname{du}{\left(u \right)} = 6 u$.

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

      1. The integral of the exponential function is itself.

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

      Now evaluate the sub-integral.

   3. Use integration by parts:

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

      Let $u{\left(u \right)} = 6 u$ and let $\operatorname{dv}{\left(u \right)} = e^{u}$.

      Then $\operatorname{du}{\left(u \right)} = 6$.

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

      1. The integral of the exponential function is itself.

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

      Now evaluate the sub-integral.

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

      $$\int 6 e^{u}\, du = 6 \int e^{u}\, du$$

      1. The integral of the exponential function is itself.

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

      So, the result is: $6 e^{u}$

   Now substitute $u$ back in:

   $$x \log{\left(x \right)}^{3} - 3 x \log{\left(x \right)}^{2} + 6 x \log{\left(x \right)} - 6 x$$

2. Now simplify:

   $$x \left(\log{\left(x \right)}^{3} - 3 \log{\left(x \right)}^{2} + 6 \log{\left(x \right)} - 6\right)$$

3. Add the constant of integration:

   $$x \left(\log{\left(x \right)}^{3} - 3 \log{\left(x \right)}^{2} + 6 \log{\left(x \right)} - 6\right)+ \mathrm{constant}$$

The answer is:

$$x \left(\log{\left(x \right)}^{3} - 3 \log{\left(x \right)}^{2} + 6 \log{\left(x \right)} - 6\right)+ \mathrm{constant}$$