The integral $\displaystyle \int \dfrac{1}{\tan^{-1}(x)}dx$ does not have an anti-derivative expressible in elementary terms, as confirmed by Wolfram Alpha (W|A). The original problem involves demonstrating that the differential equation $xdy - ydx = \tan^{-1}(y/x)dx$ can be solved using the substitution $y = vx$. This leads to a separable differential equation $x^{2}dv - \tan^{-1}(v)dx = 0$, which simplifies to $\dfrac{1}{\tan^{-1}(v)}dv = \dfrac{1}{x^2}dx$. The solution results in an implicit form, indicating that the differential equation does not yield a straightforward explicit solution.
PREREQUISITESMathematics students, educators, and professionals dealing with calculus and differential equations, particularly those interested in advanced integration techniques and their applications.
MarkFL said:That function (the integrand) does not have an anti-derivative that can be expressed in elementary terms (at least according to W|A). Is this perhaps part of another problem, where this may be the wrong result?