Integral of (80cosx)*(sin^3(x)) dx

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

   Then let $du = \cos{\left(x \right)} dx$ and substitute $80 du$:

   $$\int 80 u^{3}\, du$$

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

      $$\int u^{3}\, du = 80 \int u^{3}\, du$$

      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: $20 u^{4}$

   Now substitute $u$ back in:

   $$20 \sin^{4}{\left(x \right)}$$

2. Add the constant of integration:

   $$20 \sin^{4}{\left(x \right)}+ \mathrm{constant}$$

The answer is:

$$20 \sin^{4}{\left(x \right)}+ \mathrm{constant}$$