Discrepancy in answer for integration problem

I was given the following integral to solve:

integrate  2/(x*ln (x²)) with respect to x.

My workings are as follows:

Since d/dx ( ln (x²)) = 2x/ x² =2/x ,

∫ 2/(x*ln (x²)) dx  = ∫ (2/x)/(ln (x²)) dx

= ln ( ln (x² ) ) +C

However, the answer I am given in the worksheet is ln |ln|x||+C .

Am I wrong?

Student X

Your answer and that given in the worksheet are both correct.  Here is how you can arrive at the alternate solution:

∫ 2/(x *ln (x² ) ) dx  = ∫  2/ ( x*2 ln |x|) dx

= ∫ 1/ (x *ln |x|) dx

= ∫ (1/x) / (ln |x|)  dx

                                 = ln|ln |x|| +C  (shown)

How might you then proceed to reconcile the seemingly obvious differences in both answers?

Here is how things work:

Your answer   = ln (ln (x²) ) +C

= ln|2 ln|x|| +C

                    = ln 2 +ln|ln|x|| +C

                    = ln|ln|x||+ (C+ln 2)

With regards to the above, C+ln 2 can be further aggregated to form a new constant of integration.

(Note that in specific instances, I have replaced brackets with moduluo signs; this is to ensure the contents of the natural logarithm are positive, thereby ensuring the logarithmic function itself is properly defined.)

Hope this helps. Peace.

Best Regards,

Mr Koh