Question

Solve the following:

`int_0^1 e^(x^2)*x^3dx`

Answer 1

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)`.