SOLUTION: Perform the indicated integral. i. ∫ x^2e^xdx ii. ∫ x ln(x)dx

i. ∫ x²exdx

 u = x²    dv = exdx
du = 2xdx   v = ex

uv - ∫vdu

(x²)ex - ∫ex(2xdx)

(1)     x²ex - 2∫xexdx

Now we do ∫xexdx 

 u = x     dv = exdx
du = dx    v = ex

uv - ∫vdu   
 
xex - ∫exdx

xex - ex

Now substitute that for the integral in expression (1),
and put on the arbitrary constant + C

(1)     x²ex - 2∫xexdx
        x²ex - 2(xex - ex)
        x²ex - 2xex + 2ex + C

ii. ∫ x ln(x)dx

 u = ln(x)     dv = xdx
du = dx       v = x²

You finish.

Edwin