\int x^3e^xdx∫x3exdx
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
