One more question that i could not understand

Question

anonymous · Answer

$$\int\limits_{}^{} \coth(\frac{ \theta }{ \sqrt{3} }) d \theta $$

hartnn · Answer

write coth = cosh / sinh

hartnn · Answer

then use 
$\ \huge \text{In general,}\int \frac{f’(x)}{f(x)}\:dx=\ln|f(x)|+c 
$

hartnn · Answer

where f(x) = sinh

hartnn · Answer

making a substitution before will simplify things
x= theta / root 3 
dx=... ?

anonymous · Answer

dx= 1/root3 d(theta)

hartnn · Answer

yeah, so dtheta  = root 3 dx
rewrite the integral with this substitution

anonymous · Answer

$$\int\limits_{}^{} \frac{ \cosh }{ \sinh } x \sqrt{3}dx$$

hartnn · Answer

$\large \int\limits_{}^{} \frac{ \cosh x}{ \sinh x}  \sqrt{3}dx$

hartnn · Answer

here f(x) = sinh x
and use that general formula i gave.
what u get ?

anonymous · Answer

$$\sqrt{3}\ln \sinh +c$$

hartnn · Answer

thats correct.
did u get all steps ?

hartnn · Answer

and u missed a 'x'

hartnn · Answer

don't forget to resubstitute 
x= theta / root 3

anonymous · Answer

ok i get it now..  Thank for your guiden

hartnn · Answer

welcome ^_^

anonymous · Answer

but how come the last answer become this...$$\sqrt{3}\ln[ \frac{ e^\frac{ \theta }{ \sqrt{3} }-e^\frac{ - \theta }{ \sqrt{3} } }{ 2 }]$$

hartnn · Answer

by using the defination of sin h theta.

anonymous · Answer

ok.. using the hyperbolic fuctions >>

hartnn · Answer

yes, u know the difinition, right ?

anonymous · Answer

yes.. i know.. but i still not remember it..

hartnn · Answer

$\huge \sinh x= \frac{e^x-e^{-x}}{2}$