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Question
Evaluate `int_0^1 "e"^(x^2)*"x"^3 "d"x`
Solution
Let I = `int_0^1 "e"^(x^2)*"x"^3 "d"x`
= `int_0^1 "e"^(x^2)*x^2* x "d"x`
Put x2 = t
∴ 2x dx = dt
∴ x dx = `1/2 "dt"`
When x = 0, t = 0
When x = 1, t = 1
∴ I = `1/2 int_0^1 "e"^"t"*"t" "dt"`
= `1/2{["t"int"e"^"t" "dt"]_0^1 - int_0^1["d"/"dt"("t")int"e"^"t" "dt"]"dt"}`
= `1/2[["t"*"e"^"t"]_0^1 - int_0^1 1*"e"^"t" "dt"]`
= `1/2 [(1*"e"^1 - 0) - ["e"^"t"]_0^1]`
= `1/2 ["e" - ("e"^1 - "e"^0)]`
= `1/2 ("e" - "e" + 1)`
∴ I = `1/2`
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