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 Q 3  Integrate the functions \frac{1}{\sqrt { ( 2- x)^2+ 1 }}

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\frac{1}{\sqrt { ( 2- x)^2+ 1 }}

let suppose 2-x =t
then, -dx =dt
\Rightarrow \int\frac{1}{\sqrt{(2-x)^2+1}}dx = -\int \frac{1}{\sqrt{t^2+1}}dt

using the identity 

\int \frac{1}{\sqrt{x^2+1}}dt=log\left | x+\sqrt{x^2+1} \right |
                                                   
                              \\= -\log\left | t+\sqrt{t^2+1} \right |+C\\ =-\log\left | 2-x+\sqrt{(2-x)^2+1} \right |+C\\ =\log \left | \frac{1}{(2-x)+\sqrt{x^2-4x+5}} \right |+C

Posted by

manish

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