Integrate tanx; by parts as follows:—

i.e. 0= — 1, which is absurd.

It might be conjectured at first sight that the above result is due to a disregarded constant of integration, but this is not so, because we can imagine all the integrals taken between the same limits, in which case there is no constant. The true explanation is to be found by considering the proof of the theorem of integration by parts.