Integral of (ln(x)+x^(1/2))/x 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 $du$:

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

      1. Integrate term-by-term:

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

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

         1. Let $u = \frac{u}{2}$.

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

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

            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: $2 e^{u}$

            Now substitute $u$ back in:

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

         The result is: $\frac{u^{2}}{2} + 2 e^{\frac{u}{2}}$

      Now substitute $u$ back in:

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

   Method #2

   1. Rewrite the integrand:

      $$\frac{\sqrt{x} + \log{\left(x \right)}}{x} = \frac{\log{\left(x \right)}}{x} + \frac{1}{\sqrt{x}}$$

   2. Integrate term-by-term:

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

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

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

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

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

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

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

               $$\int \left(- u\right)\, du$$

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

                  $$\int u\, du = - \int u\, du$$

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

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

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

               Now substitute $u$ back in:

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

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

         Now substitute $u$ back in:

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

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

         $$\int \frac{1}{\sqrt{x}}\, dx = 2 \sqrt{x}$$

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

2. Add the constant of integration:

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

The answer is:

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