Question

Evaluate the following:

`int_0^1 x^3"e"^(-2x)  "d"x`

Answer 1

Bernoulli’s formula,

`int "uv"  "d"x` = uv₁ – u¹v₂ + u²v₃ – u³v₄ + ….

u = x³, v = e^–2x

u¹ = 3x², v₁ = `("e"^(-2x))/(-2)`

u² = 6, v₂ = `("e"^(-2x))/4`

u³ = 6, v₃ = `("e"^(-2x))/(-8)`

u⁴ = 0, v₃ = `("e"^(- 2x))/16`

`int_0^1 x^3 "e"^(-2x)  "d"x = [(x^3"e"^(-2))/(-2) - (3x^2"e"^(-2x))/4 + (6x "e"^(-2x))/(-8) - (6"e"^(-2x))/16]_0^1`

`int_0^1 x^3 "e"^(-2x)  "d"x = "e"^(-2x)[x^3/(- 2) - (3x^2)/4 + (6x)/(-8) -6/16]_0^1`

= `"e"^(-2)[- 1/2 - 6/4 - 3/8] - "e"^0 (- 3/8)`

= `"e"^(-2) ((-4 - 12 - 3)/8) + 3/8`

= `3/8 - 19/8 "e"^(-2)`

∴ `int_0^1 x^3"e"^(-2x)  "d"x = 3/8 - 19/(8"e"^2)`