Answer:
Option A
Explanation:
We have
I= $\int\frac{dx}{(1+x)\sqrt{8+7x-x^{2}}}$
$I=\int\frac{dx}{(1+x)\sqrt{(8-x)(1+x)}}$
$I=\int\frac{dx}{\sqrt{\frac{8-x}{1+x}(1+x)^{2}}}$
Put $\frac{8-x}{1+x}=t^{2}\Rightarrow\left(\frac{(1+x)(-1)-(8-x)}{(1+x)^{2}}\right)dx=2t dt$
$\Rightarrow$ $\frac{-9}{(1+x)^{2}}dx=2t dt$
$\therefore$ $I=\frac{-2}{9}\int\frac{tdt}{t}$
$I =\frac{-2}{9}\int dt=-\frac{2}{9}t+c$
$\therefore$ I=$-\frac{2}{9}\sqrt{\frac{8-x}{1+x}}+c$
