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प्रश्न
Integrate the following with respect to x.
x3e3x
उत्तर
= `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` |
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