Question

Integrate the following with respect to x:

x³ sin x

Answer 1

`int x^3 sin x`

u = x³

u’ = 3x²

u” = 6x 

u”’ = 6 

u’^v = 0

dv = sin x dx

⇒ v = `int sin x  "d"x`

= – cosx

v₁ = `int "v"  "d"x`

= `int - cos x  "d"x`

= `- int  cos x  "d"x`

= – sin x

v₂ = `int  "v"_1  "d"x`

= `int - sin x  "d"x`

= `- int sin x  "d"x`

= – (– cos x)

= cos x

v₃ = `int "v"_2  "d"x`

= `int cos x  "d"x`

= sin x

v₄ = `int "v"_3  "d"x`

= `int sin x  "d"x`

= – cos x

`int "u"  "dv"` = uv – u’ v₁ + u” v₂ – u”’ v₃ + u’^vv₄ – ………..

`int x^3 sin x  "d"x = x^3 (- cos x) – 3x^2 (- sin x) + 6x (cos x) – 6 sin x + 0 (- cos x) + "c"`

= – x³ cos x + 3x² sin x + 6x cos x – 6 sin x + c