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)
