Find the integral of arctan(x)

This is a complex question, the proof of which is unlikely to be asked at maths A-level, however the process behind the proof practices fundamental knowledge for the subject. If a student can successfully follow and answer my proof, they are very likely headed for an A* grade.I will denote 'integral' by "int" and multiplication by "*". A value with a ' signifies a differential.It is clear that int arctan(x) cannot be done through simple integration. The most straightforward approach is to integrate by parts.arctan(x)*1 = uv'so v'=1, thus v=int(1) dx = xand u=arctan(x), thus u'=(d/dx)(arctan(x))This step to find u' requires a new line of working, as it is also individually complex.Let y = arctan(x)Thus x = tan(y)so dx/dy = sec^2(y)Using a known trigonometric identity; 1 + tan^2(a) = sec^2(a)Thus dx/dy = 1 + tan^2(y). So (as y = arctan(x)), dx/dy = 1 + x^2And hence dy/dx = u' = 1/(1+x^2)Now we can return to our integration by parts. The answer takes the form uv - int(u'v) dx as we know.So substituting the variables in gives int(arctan(x)) = xtan(x) - int(x/(1+x^2)) dxThis is an indefinite integral of the form f'(x)/f(x)And so the final answer is (x)arctan(x) -(0.5)ln(1 + x^2) + k (do not forget the constant, k!).