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प्रश्न
Integrate the following with respect to x.
`x^5 "e"^x`
उत्तर
`int x^5 "e"^(x^2) "d"x = int x x^4 "e"^(x^2) "d"x`
Let t = x2
`"dt"/("d"x)` = 2x
dt = 2x dx
x dx = `"dt"/2`
`int x x^4 "e"^(x^2) "d"x = int "t"^2 "e"^"t" ("dt"/2)`
= `1/2 int "udv"`
= `1/2 ["uv" - "u"^"I""v"_1 + "u"^"II""v"_2 .........]`
`1/2 int "t"^2 "e"^"t" "d"x = 1/2 [("t"^2) "e"^"t" - (2"t")("e"^"t") + 2("e"^"t")]`
= `1/2 "e"^"t" ["t"^2- 2"t" + 2]+ "c"`
= `1/2 "e"^(x^2) [(x^2)^2 - 2(x^2) + 2] + "c"`
=`1/2 "e"^(x^2) [x^4 - 2x^2 + 2] + "c"`
| Successive derivatives | Repeated Integrals |
|
Take u = t2 uI = 2t uII = 2 uIII = 0 |
dv = `"e"^"t" "d"x` `int "dv" = int "e"^"t" "d"x` v = et v1 = `int "e"^"t" "dt"` = et v2 = `int "e"^"t" "dt"` = et |
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