= 1/3 x^3(lnx+ 1)- 1/3∫x^2dx

= 1/3 x^3(lnx+ 1)-x^3/9+c

(2) ∫(x^2-2x+5)e^(2x)dx

= 1/2(x^2-2x+5)e^(2x)-∫(x- 1)e^(2x)dx

= 1/2(x^2-2x+5)e^(2x)- 1/2(x- 1)e^(2x)+ 1/2∫e^(2x)dx

= 1/2(x^2-2x+5)e^(2x)- 1/2(x- 1)e^(2x)+ 1/4*e^(2x)+c

(3) The original formula =-1/2 ∫ (cos2x) 3d (cos2x) =-1/8 (cos2x) 4+c.

(4) The original formula = ∫ (tanx) 4d (tanx) =1/5 (tanx) 5+c.

(5) Let x = tantdx = (sect) 2dt.

The original formula = ∫ (sect) 2dt/(sect) 3 = ∫ Costdt = Sint+c = x/√ (1+x 2)+c.