#\int x^3e^xdx#
Applying integration by parts as: #\int uv'=uv-\int u'v#
#u=x^3,u'=3x^2,v'=e^x,v=e^x#
#=x^3e^x-\int 3x^2e^xdx#
#\int 3x^2e^xdx=3(x^2e^x-2(e^x x-e^x))# as under
=#\int 3x^2e^xdx#
taking the constant out as: #\int a\cdot f(x)dx=a\cdot \int f(x)dx#
#=3\int x^2e^xdx#
Applying integration by parts as: #\int uv'=uv-\int u'v#
#u=x^2,u'=2x,v'=e^x,v=e^x#
#=3(x^2e^x-\int2xe^xdx)#
taking the constant out as: #\int a\cdot f(x)dx=a\cdot \int f(x)dx#
#=3(x^2e^x-2\int xe^xdx)#
Applying integration by parts as: #\int uv'=uv-\int u'v#
#u=x,u'=1,v'=e^x,v=e^x#
#=3(x^2e^x-2(xe^x-\int 1e^xdx))#
#=3(x^2e^x-2(e^x x-\int e^xdx))#
Using the common integral #\inte^xdx=e^x#
#=3(x^2e^x-2(e^x x-e^x))#
#=x^3e^x-3(x^2e^x-2(e^x x-e^x))#
Adding a constant to the solution,
#=x^3e^x-3(x^2e^x-2(e^x x-e^x))# +C