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Question
`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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