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Q13  Integrate the functions\frac{1}{\sqrt { ( x-1)( x-2 )}}

Answers (1)

best_answer

(x-1)(x-2) can be also written as 
= x^2-3x+2
=(x-\frac{3}{2})^2-(\frac{1}{2})^2
 

therefore

\int \frac{1}{\sqrt{(x-1)(x-2)}}dx= \int \frac{1}{\sqrt{(x-\frac{3}{2})^2-(\frac{1}{2})^2}}dx
                                                  let suppose
                                                  x-3/2 = t \Rightarrow dx =dt
                                                   Now,          

                                                   \Rightarrow \int \frac{1}{\sqrt{(x-\frac{3}{2})^2-(\frac{1}{2})^2}}dx = \int \frac{1}{\sqrt{t^2-(\frac{1}{2})^2}}dt.............by using formula \int \frac{1}{\sqrt{x^2-a^2}}=\log\left | x+\sqrt{x^2+a^2} \right |
                                                                                                             \\= \log \left | t+\sqrt{t^2-(1/2)^2} \right |+C\\ = \log \left | (x-\frac{3}{2})+\sqrt{x^2-3x+2} \right |+C

                                                   
                                                   

Posted by

manish

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