(7) ∫x^2lnx dx
(8) ∫xcos^2(x) dx
(9) ∫x^2arctanx dx

(7) ∫x^2lnx dx= (1/3)∫lnx d(x^3)
=(1/3)[(x^3)lnx-∫x^2 dx]
=(1/9)[(x^3)(3lnx-1)+C.

(8) ∫x*cos^2(x) dx =∫x[1+cos(2x)]/2 dx
=(x^2)/4+(1/4)∫xdsin(2x)
=(x^2)/4+(1/4)[x*sin(2x)-∫sin(2x)dx]
=(x^2)/4+(1/4)[x*sin(2x)+(1/2)cos(2x)]+C.
=[2(x^2)+2x*sin(2x)+cos(2x)]/8+C.

(9) ∫x^2arctanx dx=(1/3)∫arctanx d(x^3)
=(1/3)[(x^3)*arctanx -∫x^3/(1+x^2)dx]
=(1/3)[(x^3)*arctanx -(1/2)∫[1-1/(1+x^2)d(x^2)]
=[2(x^3)*arctanx -x^2+ln(1+x^2)]/6+C.