integrate [tan(ln x) tan(ln(x/2)) tan(ln2)]/x w.r.t x

Hi Suchita,

Here I'm presuming that last term must have been tan(ln2x)

Make the substitution lnx = t,

then 1/xdx = dt.

So the integral reduces to {tant*tan(t-ln2)*tan(t+ln2) dt}

Now tan(t-ln2) = {tant - tan(ln2)}/{1+tant*tan(ln2)}

and tan(t+ln2) = {tant + tan(ln2)}/{1-tant*tan(ln2)}

Substitute it in the integral, and make the substitution tan²t = y

You will have the integral [ (y-a²) / [2{1-(a²y)}(1+y)] ]dy,      where a = ln2.

Which is a standard integral of the form, Linear/Quadratic...... which one can easily integrated by the standard method.

Hope that helps,

All the best,

Regards,

Ashwin (IIT Madras).

Approved