Integrate by parts the following function: ln(x)/x^3

Let integrate be denoted by the letter I. For instance I(f) is the integration of a function f . Then Integration by parts states that I(u v') = uv - I(u' v), where u,v are function with u', v' their respective derivatives. Applying this to the above forumla we set u= ln(x) and v' = 1/ x³, then integrating v' gives us v= -1/(2 x²) and differentiating u gives u' = 1/x. Then applying the integration by parts formula we arrive at: I( ln(x)/x³ ) = -ln(x) / (2x²) + 1/2 I(1/x³). So the problem boils down to integrating 1/x³ which is -1/(2x²). Which gives us the answer: I(ln(x)/x³) = -ln(x)/(2x²) - 1/(4x²)