The question is asking for the indefinite integral of a function (9 - 2x) ^ 12.
To get the answer, we use the power rule for integration, which states that ∫x^n dx = x^(n+1)/(n+1) + C, where C is the constant of integration.
Notice that we are dealing with a composite function of the form ∫f(g(x)) dx in which we let u = g(x). In this case, let's consider u = (9 - 2x).
But we have to take into account the chain rule! So we'll also need du/dx = -2 for our substitution. To factor this into our integral we re-arrange to get dx = du / -2.
Therefore, we rewrite the integral as ∫(u^12 * du / -2).
Applying the power rule gives us: (-1/2) * (u^13 / 13) + C.
Don't forget to substitute back 9 - 2x for u, which gives us our final answer:
4096x^13/13 - 18432x^12 + 497664x^11 - 8211456x^10 + 92378880x^9 - 748268928x^8 + 4489613568x^7 - 20203261056x^6 + 68186006064x^5 - 170465015160x^4 + 306837027288x^3 - 376572715308x^2 + 282429536481x + C.
So, the indefinite integral of (9 - 2x) ^ 12 with respect to x is given by the above polynomial expression + a constant of integration 'C'.