Integral of lgx/x^3 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 $du$:

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

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

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

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

         1. There are multiple ways to do this integral.

            Method #1

            1. Let $u = - 2 u$.

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

               $$\int \frac{e^{u}}{4}\, du$$

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

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

                  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:

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

            Method #2

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

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

               $$\int \frac{1}{4}\, du$$

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

                  $$\int \left(- \frac{1}{2}\right)\, du = - \frac{\int 1\, du}{2}$$

                  1. The integral of a constant is the constant times the variable of integration:

                     $$\int 1\, du = u$$

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

               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:

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

         1. Let $u = - 2 u$.

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

            $$\int \frac{e^{u}}{4}\, du$$

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

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

               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: $\frac{e^{- 2 u}}{4}$

      Now substitute $u$ back in:

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

   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 $\operatorname{dv}{\left(x \right)} = \frac{1}{x^{3}}$.

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

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

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

         $$\int \frac{1}{x^{3}}\, dx = - \frac{1}{2 x^{2}}$$

      Now evaluate the sub-integral.

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

      $$\int \left(- \frac{1}{2 x^{3}}\right)\, dx = - \frac{\int \frac{1}{x^{3}}\, dx}{2}$$

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

         $$\int \frac{1}{x^{3}}\, dx = - \frac{1}{2 x^{2}}$$

      So, the result is: $\frac{1}{4 x^{2}}$

2. Now simplify:

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

3. Add the constant of integration:

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

The answer is:

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