Here,
I=int(lnx)^2/x^3dxI=∫(lnx)2x3dx
=int(lnx)^2*x^(-3)dx=∫(lnx)2⋅x−3dx
"Using "color(blue)"Integration by Parts"Using Integration by Parts.
I=(lnx)^2intx^(-3)dx-int(d/(dx)((lnx)^2)intx^(-3)dx)dxI=(lnx)2∫x−3dx−∫(ddx((lnx)2)∫x−3dx)dx
=(lnx)^2(x^(-2)/(-2))-int2(lnx)*1/x(x^(-2)/(-2))dx=(lnx)2(x−2−2)−∫2(lnx)⋅1x(x−2−2)dx
=-(lnx)^2/(2x^2)+int(lnx)(x^(-3))dx=−(lnx)22x2+∫(lnx)(x−3)dx
Again, "using "color(blue)"Integration by Parts"using Integration by Parts.
I=-(lnx)^2/(2x^2)+[lnx(x^(-2)/(-2))-int1/x(x^(-2)/(-2))dx]I=−(lnx)22x2+[lnx(x−2−2)−∫1x(x−2−2)dx]
=-(lnx)^2/(2x^2)-(lnx)/(2x^2)+1/2intx^(-3)dx=−(lnx)22x2−lnx2x2+12∫x−3dx
=-(lnx)^2/(2x^2)-(lnx)/(2x^2)+1/2(x^(-2)/(-2))+c=−(lnx)22x2−lnx2x2+12(x−2−2)+c
=-(lnx)^2/(2x^2)-(lnx)/(2x^2)-1/(4x^2)+c=−(lnx)22x2−lnx2x2−14x2+c
I=-1/(4x^2)[2(lnx)^2+lnx+1]+cI=−14x2[2(lnx)2+lnx+1]+c
