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  4. ∫ (tan (ln x) tan(ln (x/2))tan (ln 2)/x)dx=

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Q. $\int \frac{tan \left(ln\, x\right) tan\left(ln \frac{x}{2}\right)tan\,\left(ln\,2\right)}{x}dx=$

Integrals

Solution:

$ln\,x=ln\left(\frac{x}{2}\right)+ln\,2$
$\Rightarrow tan\left(ln\,x\right)=\frac{tan\left(ln\,x / 2\right)+tan\left(ln\,2\right)}{1-tan\left(ln\,x/ 2\right)tan\left(ln\,2\right)}$
$\Rightarrow tan\left(ln\,x\right)tan\left(ln \frac{x}{2}\right)tan\left(ln\,2\right)=tan\left(ln\,x\right)-tan\left(ln \frac{x}{2}\right)$$-tan\left(ln\,2\right)$
$\therefore I=\int \frac{tan\left(ln\,x\right)}{x}dx-\int\frac{tan\left(ln\,x / 2\right)}{x}dx-\int\frac{tan\left(ln\,2\right)}{x}dx$
$=ln\,sec\left(ln\,x\right)-ln\,sec\left(ln \frac{x}{2}\right)-tan\left(ln\,2\right)ln\,x$
$=ln\left\{\frac{sec\left(ln\,x\right)}{sec\left(ln\left(x / 2\right)\right)x^{tan\,ln\,2}}\right\}+C$