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प्रश्न
Integrate the following with respect to x:
x5ex2
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उत्तर
`int x^5 "e"^(x^2) "d"x = int (x2)^2 * "e"^(x^2) * x "d"x`
Put x2 = t
2x dx = dt
⇒ x dx = `1/2 "dt"`
`int x^5 "e"^(x^2) "d"x = int "t"^2 * "e"^"t" * 1/2 "dt"`
= `1/2 int "t"^2 "e"^"t" "dt"` .........(1)
Consider `int "t"^2 "e"^"t" "dt"`
u = t2
u' = 2t
u" = 2
u"' = 0
dv = et dt
⇒ v = `int "e"^"t" "dt"`
⇒ v = et
v1 = `int "v" "dt"`
= `int "e"^"t" "dt"`
= et
v2 = `int "v"_1 "dt"`
= `int "e"^"t" "dt"`
= et
v3 = `int "v"_2 "dt"`
= `int "e"^"t" "dt"`
= et
`int u" "dv"` = uv – u'v1 + u"v2 – u"'v3 + u"'v4 – ...........
`int "t"^2 "e"^"t" "dt"` = t2 et – 2t et + 2et + 0et
= `("t"^2 - 2"t" + 2) "e"^"t" + "c"`
Substtuting in equation (1) we get
`int x^5 "e"^(x^2) "d"x = 1/2["t"^2 - 2"t" + 2] "e"^"t" + "c"`
= `1/2 [(x^2)^2 - 2x^2 + 2] "e"^(x^2) + "c"`
`int x^5 "e"^(x^2) "d"x = 1/2 [x^4 - 2x^2 + 2] "e"^(x^2) + "c"`
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