Integral of (2x+5)e^3xdx dx

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

   $$\int \left(2 x + 5\right) e^{3} x 1\, dx = e^{3} \int x \left(2 x + 5\right)\, dx$$

   1. Rewrite the integrand:

      $$x \left(2 x + 5\right) = 2 x^{2} + 5 x$$

   2. Integrate term-by-term:

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

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

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

            $$\int x^{2}\, dx = \frac{x^{3}}{3}$$

         So, the result is: $\frac{2 x^{3}}{3}$

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

         $$\int 5 x\, dx = 5 \int x\, dx$$

         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: $\frac{5 x^{2}}{2}$

      The result is: $\frac{2 x^{3}}{3} + \frac{5 x^{2}}{2}$

   So, the result is: $\left(\frac{2 x^{3}}{3} + \frac{5 x^{2}}{2}\right) e^{3}$

2. Now simplify:

   $$\frac{x^{2} \cdot \left(4 x + 15\right) e^{3}}{6}$$

3. Add the constant of integration:

   $$\frac{x^{2} \cdot \left(4 x + 15\right) e^{3}}{6}+ \mathrm{constant}$$

The answer is:

$$\frac{x^{2} \cdot \left(4 x + 15\right) e^{3}}{6}+ \mathrm{constant}$$