Integral of y=3sin²x×cosxdx dx

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

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

   1. Let $u = \sin{\left(x \right)}$.

      Then let $du = \cos{\left(x \right)} dx$ and substitute $du$:

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

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

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

      Now substitute $u$ back in:

      $$\frac{\sin^{3}{\left(x \right)}}{3}$$

   So, the result is: $\sin^{3}{\left(x \right)}$

2. Add the constant of integration:

   $$\sin^{3}{\left(x \right)}+ \mathrm{constant}$$

The answer is:

$$\sin^{3}{\left(x \right)}+ \mathrm{constant}$$