Integral of y^2e^xy^2+6x dx

1. Integrate term-by-term:

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

      $$\int 6 x\, dx = 6 \int x\, dx$$

      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: $3 x^{2}$

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

      $$\int y^{2} y^{2} e^{x}\, dx = y^{4} \int e^{x}\, dx$$

      1. The integral of the exponential function is itself.

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

      So, the result is: $y^{4} e^{x}$

   The result is: $3 x^{2} + y^{4} e^{x}$

2. Add the constant of integration:

   $$3 x^{2} + y^{4} e^{x}+ \mathrm{constant}$$

The answer is:

$$3 x^{2} + y^{4} e^{x}+ \mathrm{constant}$$