Integration?

Let u = x and dv = (2 - 3x)¹¹ dx
Then, du = dx and v = (-1/3) (1/12) (2 - 3x)¹² = (-1/36) (2 - 3x)¹²

∫ x (2 - 3x)¹¹ dx
= u dv
= uv - ∫ v du
= x • (-1/36) (2 - 3x)¹² - ∫ (-1/36) (2 - 3x)¹² dx
= (-1/36) x (2 - 3x)¹² - (-1/36) (-1/3) (1/13) (2 - 3x)¹³ + C
= (-1/36) x (2 - 3x)¹² - (1/1404) (2 - 3x)¹³ + C
= -(1/1404) (2 - 3x)¹² [39x + (2 - 3x)] + C
= -(1/1404) (2 - 3x)¹² (2 + 36x) + C

∫ x(2-3x)^11 dx

Let u=2-3x
du = -3dx
dx = -(1/3) du
3x = 2-u
x= (2-u)/3

∫ x (2-3x)^11 dx = -(1/3) ∫ (1/3) (2-u) u^11 du
= -(1/9) ∫ (2u^11-u^12) du
= (-2/9) ∫ u^11 du + (1/9) ∫ u^12 du
=(-2/9)(1/12)u^12 + (1/9)(1/13) u^13

= (-1/54) u^12 + (1/117) u^13

replace u by 2-3x
= (-1/54) (2-3x)^12 + (1/117) (2-3x)^13 + C