Making this substitution using16 sin2(x) cos3(x) dxgives us 16 sin2(x) (1 − sin2(x)) cos(x) dx = 16 sin2(x) cos(x) dx −  cos(x)  dx.

It seems like you've made a statement rather than asking a question. However, I can see that you're working on an integral involving trigonometric functions.

You've started with the integral of 16sin^2(x)cos^3(x) dx and made a substitution to express cos^3(x) as (1 - sin^2(x))cos(x). This gives you the integral of 16sin^2(x)cos(x) dx - 16sin^2(x)cos(x) dx.

If you want to proceed with this integral, you could use the power-reducing identity for sin^2(x) = 1/2(1 - cos(2x)) and then use the substitution u = cos(x), du = -sin(x) dx for the integral.

If you need further help, please provide more details or ask a specific question.