Integral of 3xln(x+2) dx

1. Use integration by parts:

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

   Let $u{\left(x \right)} = \log{\left(x + 2 \right)}$ and let $\operatorname{dv}{\left(x \right)} = 3 x$.

   Then $\operatorname{du}{\left(x \right)} = \frac{1}{x + 2}$.

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

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

      $$\int 3 x\, dx = 3 \int x\, dx$$

      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: $\frac{3 x^{2}}{2}$

   Now evaluate the sub-integral.

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

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

   1. Rewrite the integrand:

      $$\frac{x^{2}}{x + 2} = x - 2 + \frac{4}{x + 2}$$

   2. Integrate term-by-term:

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

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

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

         $$\int \left(-2\right)\, dx = - 2 x$$

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

         $$\int \frac{4}{x + 2}\, dx = 4 \int \frac{1}{x + 2}\, dx$$

         1. Let $u = x + 2$.

            Then let $du = dx$ and substitute $du$:

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

            1. The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$.

            Now substitute $u$ back in:

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

         So, the result is: $4 \log{\left(x + 2 \right)}$

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

   So, the result is: $\frac{3 x^{2}}{4} - 3 x + 6 \log{\left(x + 2 \right)}$

3. Now simplify:

   $$\frac{3 x^{2} \log{\left(x + 2 \right)}}{2} - \frac{3 x^{2}}{4} + 3 x - 6 \log{\left(x + 2 \right)}$$

4. Add the constant of integration:

   $$\frac{3 x^{2} \log{\left(x + 2 \right)}}{2} - \frac{3 x^{2}}{4} + 3 x - 6 \log{\left(x + 2 \right)}+ \mathrm{constant}$$

The answer is:

$$\frac{3 x^{2} \log{\left(x + 2 \right)}}{2} - \frac{3 x^{2}}{4} + 3 x - 6 \log{\left(x + 2 \right)}+ \mathrm{constant}$$