intx^2expxsinxx01

Using symbolic integration, we have:

∫x^2*exp(x)*sin(x) dx = (x^2 - 2x + 2)*exp(x)*sin(x) - (x^3 - 3x^2 + 6x - 6)*exp(x)*cos(x) + C

Therefore,

int(x^2*exp(x)sin(x),x,0,1) = [(1^2 - 21 + 2)exp(1)sin(1) - (1^3 - 31^2 + 61 - 6)*exp(1)cos(1)] - [(0^2 - 20 + 2)exp(0)sin(0) - (0^3 - 30^2 + 60 - 6)*exp(0)*cos(0)]

= [-1.1504] - [-6]

= 4.8496

Therefore, the definite integral of x^2*exp(x)*sin(x) from 0 to 1 is approximately 4.8496.