I = int udv = uv - int vduI=∫udv=uv−∫vdu
u=ln(3x-x^2) => du=(3-2x)/(3x-x^2)dxu=ln(3x−x2)⇒du=3−2x3x−x2dx
dv = xdx => v = int xdx = x^2/2dv=xdx⇒v=∫xdx=x22
I = x^2/2ln(3x-x^2) - int x^2/2 (3-2x)/(3x-x^2)dxI=x22ln(3x−x2)−∫x223−2x3x−x2dx
I = 1/2[x^2ln(3x-x^2) - int (x^2(2x-3))/(x^2-3x)dx]I=12[x2ln(3x−x2)−∫x2(2x−3)x2−3xdx]
I_1 = int (2x^3-3x^2)/(x^2-3x)dx = int (2x^3-6x^2+3x^2-9x+9x)/(x^2-3x)dxI1=∫2x3−3x2x2−3xdx=∫2x3−6x2+3x2−9x+9xx2−3xdx
I_1 = int (2x(x^2-3x)+3(x^2-3x)+9x)/(x^2-3x)dxI1=∫2x(x2−3x)+3(x2−3x)+9xx2−3xdx
I_1 = int (2x+3+(9x)/(x(x-3)))dx = int (2x+3+9/(x-3))dxI1=∫(2x+3+9xx(x−3))dx=∫(2x+3+9x−3)dx
I_1 = x^2 + 3x + 9ln|x-3|+CI1=x2+3x+9ln|x−3|+C
I = 1/2[x^2ln(3x-x^2) - x^2-3x-9ln|x-3|]|_a^(a^2)I=12[x2ln(3x−x2)−x2−3x−9ln|x−3|]∣∣∣a2a
I = 1/2(a^4ln(3a^2-a^4)-a^4-3a^2-9ln|a^2-3|-a^2ln(3a-a^2)+a^2+3a+9ln|a-3|)I=12(a4ln(3a2−a4)−a4−3a2−9ln∣∣a2−3∣∣−a2ln(3a−a2)+a2+3a+9ln|a−3|)
I = 1/2(3a-2a^2-a^4+9ln|a-3|/|a^2-3|+a^2ln((3a^2-a^4)^(a^2)/(3a-a^2)))I=12⎛⎝3a−2a2−a4+9ln|a−3||a2−3|+a2ln⎛⎝(3a2−a4)a23a−a2⎞⎠⎞⎠