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

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

Grade:12th Pass

2 Answers

Ashwin Muralidharan IIT Madras
290 Points
10 years ago

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,


Ashwin (IIT Madras).

13 Points
3 years ago
After, ln x=t
            We had tant*tan(t-ln2)*tan(ln 2) dt
 Since: t-(t - ln 2) =ln 2
            { tan t -tan (t-ln 2)}/1+tan t*tan (t-ln2) =tanln2
            Now we have,
           tan t*tan(t-ln2)*tan(ln2)=tan t -tan(t-ln2)-tanln2.
Thus now integrate this value achieved

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