Question

Integrate the following with respect to x.

x³e^3x

Answer 1

= `int x^3 "e"^(3x)  "d"x`

= `int "udv"` = uv – u¹v₁ + u¹¹v₂ – u¹¹¹v₃ ………

`int x^3"e"^(3x)  "d"x = (x^3)(("e"^(3x))/3) - (3x^2) ("e"^(3x)/9) + 6x ("e"^(3x)/27) - +("e"^(3x)/81) + "c"`

=  `"e"^(3x) [x^3/3 - (3x^2)/9 + (6x)/27 - 6/81] + "c"`

= `"e"^(3x) [x^3/3 - x^2/3 + (2x)/9 - 2/27] + "c"`

Successive derivatives	Repeated Integrals
Take  u = x3 uI = 3x2 uII = 6x uIII = 6 uIV = 0	and dv = e3x  `int "dv" = int "e"^(3x)  "d"x` v = `"e"^(3x)/3` v1 = `"e"^(3x)/9` v2 = `"e"^(3x)/27` v3 = `"e"^(3x)/81`