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Q22  Choose the correct answer \int \frac{x dx }{( x-1)(x-2) } \: \: equals

         A ) \log |\frac{(x-1)^2}{x-2}| + C \\\\ B) \log |\frac{(x-2)^2}{x-1}| + C \\\\ C ) \log |(\frac{x-1}{x-2})^2| + C \\\\ D ) \log |{(x-1)^2}({x-2})| + C

Answers (1)

best_answer

Given integral  \int \frac{x dx }{( x-1)(x-2) }

Partial fraction of above equation,

\frac{x}{(x-1)(x-2)} = \frac{A}{(x-1)}+\frac{B}{(x-2)}

x= A(x+2)+B(x-1)                                            ..............(1)                            

Now, substituting x = 1\ and\ x = 2 in equation (1), we get

A = -1\ and\ B=2

\therefore \frac{x}{(x-1)(x-2)} = -\frac{1}{(x-1)}+\frac{2}{(x-2)}

\implies \int \frac{x}{(x-1)(x-2)}dx = \int \left \{ \frac{-1}{(x-1)}+\frac{2}{(x-2)} \right \}dx

= -\log|x-1| +2log|x-2| +C

=\log \left | \frac{(x-2)^2}{x-1} \right | +C

Therefore, the correct answer is B.

 

Posted by

Divya Prakash Singh

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