Evaluate the following definite integral: ∫ (xe^(-x^2))dx

Evaluate the following definite integral:

∫ (xe^(-x^2))dx
Note: ∫(a=0, b=1)

Any and all help is much appreciated! Thanks so much in advance!

∫ (xe^(-x^2))dx from 0 to 1

let x^2 = t, when x = 0, t = 0 and when x = 1, t = 1
2x dx = dt
x dx = dt/2

= 1/2∫ e^(-t))dt

= -(1/2)e^(-t) from [ 0 to 1]

= -1/2[1/e - 1 ]

= 1/2[1 - 1/e]

= 0.316

        w = -x²
   dw ⁄ dx = -2x
    - dw ⁄ 2 = x • dx


 b
 ∫ x • e^(-x²) • dx = (-½) • ∫ eʷ • dw = (-½) • eʷ + C
 a

          = (-½) • e^(-x²) + C eval [a→b]

          = (-½) • [ e^(-b²) − -e^(-a²) ]

          = - [ e^(-b²) + e^(-a²) ] ⁄ 2