Use the substition u = cos(x) to find the indefinite integral of -12sin(x)cos^3(x) dx

We are given the substitution to use, so the first step is to differentiate "u" with respect to x. 

du/dx = -sin(x)

Now, to replace the "dx" in the original integrand with something in terms of "du", we rearrange the differential:

dx = -1/sin(x) du

We substitute this into the original expression we are integrating; this gives: 

S -12sin(x)cos³(x) (-1/sin(x)) du

Let's do some simplifying here; the negative signs cancel, and so does sin(x):

S 12cos³(x) du

Now, simplify again using u=cos(x); this gives:

S 12u³ du

This is a simple C1-level integration; integrating with respect to "u" and adding a constant of integration, we get:

3u⁴ + c

For our final answer, replace "u" with cos(x):

3cos⁴x) + c