question_answer

If \[\int{\frac{1}{(\sin \,\,x+4)(\sin \,\,x+1)}\,\,dx=\frac{A}{\tan \frac{x}{2}-1}+B\,\,{{\tan }^{-1}}[f\,\,(x)]+c,}\]then

A)  \[A=\frac{1}{5},\] \[B=\frac{-2}{5\sqrt{15}},\] \[f\,\,(x)=\frac{4\,\,\tan x-3}{\sqrt{15}}\]

B)  \[A=\frac{1}{5},\] \[B=\frac{1}{\sqrt{15}},\] \[f\,\,(x)=\frac{4\,\,\tan \left( \frac{x}{2} \right)+1}{\sqrt{15}}\]

C)  \[A=\frac{2}{5},\] \[B=\frac{-2}{5},\] \[f\,\,(x)=\frac{4\,\,\tan x+1}{5}\]

D)  \[A=\frac{2}{5},\] \[B=\frac{-2}{5\sqrt{15}},\]\[f\,\,(x)=\frac{4\,\,\tan \frac{x}{2}+1}{\sqrt{15}}\]

E)   None of these