Question

Integrate the following with respect to x:

x² cos x

Answer 1

`int x^2 cos x  "d"x`

u = x²

u' = 2x

u' = 2

u"' 0

dv = cos x dx

⇒ v= `int cos x  "d"x`

= sin x

v₁ = `int "v"  "d"x`

= `int sinx  "d"x`

= `- cos x`

v₂ = `int "v"_1  "d"x`

= `int - cosx  "d"x`

= `- sin x`

v₃ = `int "v"_2  "d"x`

= `int- sin x  "d"x`

=  `- int  sinx  "d"x`

= `- (cos x)`

`int "u"  "dv"` = uv – u'v₁ + uv₂ – u"'v₃ + ...........

`int x^2 cos x  "d"x = x^2 sin x - 2x xx cos x + 2 xx - sin x - 0 xx cos x + "c"`

`int x^2 cos x  "d"x = x^2 sin x + 2x cosx - 2 sinx + "c"`