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Q1  Integrate the rational functions \frac{x }{( x +1)( x+2)}

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Given function \frac{x }{( x +1)( x+2)}

Partial function of this function:

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

\implies x = A(x+2)+B(x+1)

Now, equating the coefficients of x and constant term, we obtain

A+B =1

2A+B =0

On solving, 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 \frac{-1}{(x+1)}+\frac{2}{(x+2)} dx

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

=\log(x+2)^2-\log|x+1|+C

=\log\frac{(x+2)^2}{(x+1)}+C

 

 

 

 

 

Posted by

Divya Prakash Singh

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