#### Need solution for RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Excercise 18.32 Question 8

Answer : -       $-\frac{1}{\sqrt{2}} \log \left|\left(\frac{1}{x-1}+\frac{1}{2}\right)+\sqrt{\left(\frac{1}{x-1}+\frac{1}{2}\right)^{2}+\frac{1}{4}}\right|+C$

Hint :-                Use substitution and special integration formula.

Given :-                   $\int \frac{1}{\left ( x-1 \right )\sqrt{x^{2}+1}}dx$

Sol : -             Let    $I = \int \frac{1}{\left ( x-1 \right )\sqrt{x^{2}+1}}dx$

Put, $x -1 = \frac{1}{t}$

$\Rightarrow dx=-\frac{1}{t^{2}}dt$   then,

\begin{aligned} &\begin{aligned} \mathrm{I} &=\int \frac{1}{\frac{1}{t} \sqrt{\left(\frac{1}{t}+1\right)^{2}+1}}\left(-\frac{1}{t^{2}}\right) d t \\ &=-\int \frac{\frac{1}{t^{2}}}{\frac{1}{t} \sqrt{\frac{1}{t^{2}}+1+\frac{3}{t}+1}} d t\left[\because(a+b)^{2}=a^{2}+b^{2}+2 a b\right] \end{aligned} \end{aligned}

\begin{aligned} &=-\int \frac{\frac{1}{t^{2}}}{\frac{1}{t}\sqrt{\frac{1+t^{2}+2t+t^{2}}{t^{2}}}}dt \\ &=-\int \frac{\frac{1}{t^{2}}}{\frac{11}{t} \cdot \frac{1}{t} \sqrt{2 t^{2}+2 t+1}} d t \\ &=-\int \frac{\frac{1}{t^{2}}}{\frac{1}{t^{2}} \sqrt{2\left(t^{2}+t+\frac{1}{2}\right)}} d t \end{aligned}

\begin{aligned} &=-\int \frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{t^{2}+2 \cdot t \cdot \frac{1}{2}+\left(\frac{1}{2}\right)^{2}-\left(\frac{1}{2}\right)^{2}+\frac{1}{2}}} d t \\ &=-\frac{1}{\sqrt{2}} \int \frac{1}{\sqrt{\left(t+\frac{1}{2}\right)^{2}-\frac{1}{4}+\frac{1}{2}}} d t&\quad\left[\because a^{2}+b^{2}+2 a b=(a+b)^{2}\right] \\ \end{aligned}

\begin{aligned} &=-\frac{1}{\sqrt{2}} \int \frac{1}{\sqrt{\left(t+\frac{1}{2}\right)^{2}-\left(\frac{1-2}{4}\right)}} d t \\ &=-\frac{1}{\sqrt{2}} \int \frac{1}{\sqrt{\left(t+\frac{1}{2}\right)^{2}-\left(\frac{-1}{4}\right)}} d t \\ &=-\frac{1}{\sqrt{2}} \int \frac{1}{\sqrt{\left(t+\frac{1}{2}\right)^{2}+\frac{1}{4}}} d t \\ &=-\frac{1}{\sqrt{2}} \int \frac{1}{\sqrt{\left(t+\frac{1}{2}\right)^{2}+\left(\frac{1}{2}\right)^{2}}} d t \end{aligned}
Put, $t +\frac{1}{2} = u$

$\Rightarrow dt=du$ than

\begin{aligned} &\mathrm{I}=-\frac{1}{2} \int \frac{1}{\sqrt{u^{2}+\left(\frac{1}{2}\right)^{2}}} \mathrm{du} \\ &\mathrm{I}=-\frac{1}{\sqrt{2}} \log \left|u+\sqrt{u^{2}+\left(\frac{1}{2}\right)^{2}}\right|+\mathrm{c} \quad\left[\because \int \frac{1}{\sqrt{x^{2}+a^{2}}} d x=\log \left|x+\sqrt{x^{2}+a^{2}}\right|+\mathrm{c}\right] \end{aligned}

\begin{aligned} &\mathrm{I}=-\frac{1}{\sqrt{2}} \log \left|\left(t+\frac{1}{2}\right)+\sqrt{\left(t+\frac{1}{2}\right)^{2}+\frac{1}{4}}\right|+\mathrm{c} \quad\left(\because \mathrm{u}=\mathrm{t}+\frac{1}{2}\right) \\ &\mathrm{I}=-\frac{1}{\sqrt{2}} \log \left|\left(\frac{1}{x-1}+\frac{1}{2}\right)+\sqrt{\left(\frac{1}{x-1}+\frac{1}{2}\right)^{2}+\frac{1}{4}}\right|+\mathrm{c} \quad\left(\because(\mathrm{x}-1)=\frac{1}{t}\right) \end{aligned}