Question

Evaluate `int_0^1 "e"^(x^2)*"x"^3  "d"x`

Answer 1

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`