Integral of 0.5+x*lg(x) 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 $du$:

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

         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. Let $u = 2 u$.

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

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

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

               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:

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

            1. Let $u = 2 u$.

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

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

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

               Now substitute $u$ back in:

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

            So, the result is: $\frac{e^{2 u}}{4}$

         Now substitute $u$ back in:

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

      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 $\operatorname{dv}{\left(x \right)} = x$.

         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 $n \neq -1$:

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

         Now evaluate the sub-integral.

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

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

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

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

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

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

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

   The result is: $\frac{x^{2} \log{\left(x \right)}}{2} - \frac{x^{2}}{4} + \frac{x}{2}$

2. Now simplify:

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

3. Add the constant of integration:

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

The answer is:

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