Integral of 3sqrtx^2 dx

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

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

   1. Let $u = \sqrt{x}$.

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

      $$\int 2 u^{3}\, du$$

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

         $$\int u^{3}\, du = 2 \int u^{3}\, du$$

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

            $$\int u^{3}\, du = \frac{u^{4}}{4}$$

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

      Now substitute $u$ back in:

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

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

2. Add the constant of integration:

   $$\frac{3 x^{2}}{2}+ \mathrm{constant}$$

The answer is:

$$\frac{3 x^{2}}{2}+ \mathrm{constant}$$