Integral of lnx/x×✓lnx+2 dx

1. Integrate term-by-term:

   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 $du$:

         $$\int u^{\frac{3}{2}}\, du$$

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

            $$\int u^{\frac{3}{2}}\, du = \frac{2 u^{\frac{5}{2}}}{5}$$

         Now substitute $u$ back in:

         $$\frac{2 \log{\left(x \right)}^{\frac{5}{2}}}{5}$$

      Method #2

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

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

         $$\int \left(- \frac{\log{\left(\frac{1}{u} \right)}^{\frac{3}{2}}}{u}\right)\, du$$

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

            $$\int \frac{\log{\left(\frac{1}{u} \right)}^{\frac{3}{2}}}{u}\, du = - \int \frac{\log{\left(\frac{1}{u} \right)}^{\frac{3}{2}}}{u}\, du$$

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

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

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

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

                  $$\int u^{\frac{3}{2}}\, du = - \int u^{\frac{3}{2}}\, du$$

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

                     $$\int u^{\frac{3}{2}}\, du = \frac{2 u^{\frac{5}{2}}}{5}$$

                  So, the result is: $- \frac{2 u^{\frac{5}{2}}}{5}$

               Now substitute $u$ back in:

               $$- \frac{2 \log{\left(\frac{1}{u} \right)}^{\frac{5}{2}}}{5}$$

            So, the result is: $\frac{2 \log{\left(\frac{1}{u} \right)}^{\frac{5}{2}}}{5}$

         Now substitute $u$ back in:

         $$\frac{2 \log{\left(x \right)}^{\frac{5}{2}}}{5}$$

      Method #3

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

         Then let $du = \frac{dx}{2 x \sqrt{\log{\left(x \right)}}}$ and substitute $2 du$:

         $$\int 2 u^{4}\, du$$

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

            $$\int u^{4}\, du = 2 \int u^{4}\, du$$

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

               $$\int u^{4}\, du = \frac{u^{5}}{5}$$

            So, the result is: $\frac{2 u^{5}}{5}$

         Now substitute $u$ back in:

         $$\frac{2 \log{\left(x \right)}^{\frac{5}{2}}}{5}$$

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

      $$\int 2\, dx = 2 x$$

   The result is: $2 x + \frac{2 \log{\left(x \right)}^{\frac{5}{2}}}{5}$

2. Add the constant of integration:

   $$2 x + \frac{2 \log{\left(x \right)}^{\frac{5}{2}}}{5}+ \mathrm{constant}$$

The answer is:

$$2 x + \frac{2 \log{\left(x \right)}^{\frac{5}{2}}}{5}+ \mathrm{constant}$$