Answered

if in tan h inversex instead of x we have 3x-2 then what will the formula be ?

Answer :

rational
Recall the identity
[tex]\tanh^{-1}(\clubsuit)=\frac{1}{2}\ln \frac{\clubsuit+1}{\clubsuit-1}[/tex]

Therefore
[tex]\tanh^{-1}(3x-2)[/tex]

[tex]=\frac{1}{2}\ln \frac{3x-2+1}{3x-2-1}[/tex]

[tex]=\frac{1}{2}\ln \frac{3x-1}{3x-3}[/tex]
kvnmurty
The formula  for  tanh⁻¹ x:

 [tex]tanh^{-1}x=\frac{1}{2} Ln\ [\frac{1+x}{1-x}],\ \ \ | x | < 1\\\\we\ have\ |3x-2| < 1\\3x-2\ < 1,\ \ \ = >\ \ \ \ x < 1\\3x-2 > -1,\ \ \ = >\ \ \ \ x > \frac{1}{3}\\\\tanh^{-1}(3x-2)\\\\=\frac{1}{2}\ Ln\ [ \frac{1+3x-2}{1-(3x-2)} ]\\\\=\frac{1}{2}\ Ln\ [ \frac{3x-1}{3-3x} ],\ \ \ \frac{1}{3} < x < 1\\\\=\frac{1}{2}\ Ln\ [\frac{3x-1}{1-x}]-\frac{1}{2}\ Ln\ 3,\ \ \ \frac{1}{3} < x < 1\\\\OR,\\\\\frac{1}{2}\ Ln\ [ \frac{3(x-1)+2}{3(1-x)} ]\\\\=\frac{1}{2}\ Ln\ [ \frac{2}{3(1-x)} - 1 ][/tex]


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