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Q7  Integrate the functions \frac{x-1 }{\sqrt { x^2 -1 }}

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We can write above eq as
\frac{x-1 }{\sqrt { x^2 -1 }}=\int \frac{x}{\sqrt{x^2-1}}dx-\int \frac{1}{\sqrt{x^2-1}}dx............................................(i)
                   

for    \int \frac{x}{\sqrt{x^2-1}}dx                let x^2-1 = t \Rightarrow 2xdx =dt
 

\therefore \int \frac{x}{\sqrt{x^2-1}}dx=\frac{1}{2}\int \frac{dt}{\sqrt{t}}
                                     \\=\frac{1}{2}\int t^{1/2}dt\\ =\frac{1}{2}[2t^{1/2}]\\ =\sqrt{t}\\ =\sqrt{x^2-1}
Now, by using eq (i)
=\int \frac{x}{\sqrt{x^2-1}}dx-\int \frac{1}{\sqrt{x^2-1}}dx
\\=\sqrt{x^2-1}-\int \frac{1}{\sqrt{x^2}-1}dx\\ =\sqrt{x^2-1}-\log\left | x+\sqrt{x^2-1} \right |+C

                                      

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manish

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