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प्रश्न
Solve the following:
`int_0^1 e^(x^2)*x^3dx`
उत्तर
Let I = `int_0^1 e^(x^2)*x^3dx`
= `int_0^1 e^(x^2)*x^2*xdx`
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"*"tdt"`
= `(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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