∫x3ex2dx - Mathematics and Statistics

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Sum

`int x^3"e"^(x^2) "d"x`

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Solution

Let I = `int x^3*"e"^(x^2) "d"x`

= `int x^2*x"e"^(x^2) "d"x`

Put x2 = t

∴ 2x.dx = dt

∴ x dx = `"dt"/2`

∴ I = `1/2 int"te"^"t" "dt"`

= `1/2 ["t" int"e"^"t" "dt" - int["d"/"dt"("t") int"e"^"t""dt"]"dt"]`

= `1/2 ["te"^"t" - int1*"e"^"t""dt"]`

= `1/2 ("te"^"t" - "e"^"t") + "c"`

= `1/2 "e"^"t" ("t" - 1) + "c"`

∴ I = `1/2 "e"^(x^2) (x^2 - 1) + "c"`

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Chapter 1.5: Integration - Q.5

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