Integral of 6xsqrt(5-x^2) dx

1. Let $u = 5 - x^{2}$.

   Then let $du = - 2 x dx$ and substitute $- 3 du$:

   $$\int \left(- 3 \sqrt{u}\right)\, du$$

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

      $$\int \sqrt{u}\, du = - 3 \int \sqrt{u}\, du$$

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

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

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

   Now substitute $u$ back in:

   $$- 2 \left(5 - x^{2}\right)^{\frac{3}{2}}$$

2. Add the constant of integration:

   $$- 2 \left(5 - x^{2}\right)^{\frac{3}{2}}+ \mathrm{constant}$$

The answer is:

$$- 2 \left(5 - x^{2}\right)^{\frac{3}{2}}+ \mathrm{constant}$$