Use integration by parts to find the value of the indefinite integral (1/x^3)lnx ; integration with respect to dx

Let f denote an integral sign, I will write the integrand in square brackets. The formula for integration by parts is given by:f [(u)(dv/dx)]dx = uv - f [((du/dx)(v)]dxTo apply this rule we imagine our integrand ("thing to be integrated") has two parts - each are a function of x. We assign one to the variable u and differentiate it and the other part to the variable dv/dx and integrate it for v. In our given integrand the two parts are 1/x^3 and lnx. Integrating lnx will very likely not simplify what we are trying to solve (it will just give us an expression containing lnx) - so set u = lnx and differentiate it to obtain du/dx = 1/x. So dv/dx = 1/x^3 which straightforwardly gives us v = -1/(2x^2). Substitute these values into the by parts formula to obtain our integral equal to (-1/(2x^2))lnx - f [(1/x)(-1/2x^2)]dx = (-1/2x^2)lnx + f [1/(2x^3)]dx then this is easy to integrate to obtain(-1/2x^2)lnx - 1/(4x^2) +