Integral of x+2/(x+3)^3 dx

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

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

      1. Don't know the steps in finding this integral.

         But the integral is

         $$- \frac{1}{2 x^{2} + 12 x + 18}$$

      So, the result is: $- \frac{2}{2 x^{2} + 12 x + 18}$

   The result is: $\frac{x^{2}}{2} - \frac{2}{2 x^{2} + 12 x + 18}$

2. Now simplify:

   $$\frac{x^{2} \left(x^{2} + 6 x + 9\right) - 2}{2 \left(x^{2} + 6 x + 9\right)}$$

3. Add the constant of integration:

   $$\frac{x^{2} \left(x^{2} + 6 x + 9\right) - 2}{2 \left(x^{2} + 6 x + 9\right)}+ \mathrm{constant}$$

The answer is:

$$\frac{x^{2} \left(x^{2} + 6 x + 9\right) - 2}{2 \left(x^{2} + 6 x + 9\right)}+ \mathrm{constant}$$