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Question
Integrate the following with respect to x:
27x2e3x
Solution
`int 27x^2"e"^(3x) "d"x = 27 int x^2 "e"^(3x) "d"x`
u = x2
u'= 2x
u" = 2
u"' = 0
v = `"e"^(3x) * "d"x`
⇒ v = `int "e"^(3x) * "d"x`
= `("e"^(3x))/3`
v1 = `int "v" "d"x`
= `int ("e"^(3x))/3 "d"x`
= `1/3 xx ("e"^3x)/3`
= `("e"^(3x))/3^2`
v2 = `int "v"_1 "d"x`
= `int ("e"^(3x))/3^2 "d"x`
= `1/3^2 xx ("e"^3x)/3`
= `1/3^3 "e"^(3x)`
v3 = `int "v"_2 "d"x`
= `int 1/3^3 "e"^(3x) "d"x`
= `1/3^4 "e"^(3x)`
`int "u" "dv"` = uv – u'v1 + u"v2 – u"'v3 + ..........
`27 int x^2 "e"^(3x) "d"x = 27[x^2 "e"^(3x)/3 - 2x "e"^(3x)/3^2 + 2 xx "e"^(3x)/3^3 - 0 xx "e"^(3x)/3^4] + "c"`
= `9x^2 "e"^(3x) - 6x "e"^(3x) + 2"e"^(3x) + 0 + "c"`
`27int x^2 "e"^(3x) "d"x = (9x^2 - 6x + 2)"e"^(3x) + "c"`
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