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Evaluate : int (x)/((x-2)(x-1)^(2))dx...

Evaluate : `int (x)/((x-2)(x-1)^(2))dx`

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Verified by Experts

`"Let "(x)/((x-2)(x-1)^(2)) `
` =(A)/(x-2)+(B)/(x-1)+(C)/((x-1)^(2))`
` =(A(x-1)^(2)+B(x-1)(x-2)+C(x-2))/((x-2)(x-1)^(2))`
comparing the coefficients of same power
`A+B =0`
`-2A -3B +C =1`
` A+ 2B -2C =0`
on solving A=2,B =-2,C=-1
`:. int (x)/((x-2))dx - int (2)/(x-1) dx - int (1)/((x-1)^(2) ) dx`
` =2 log |x-2|-2log | x-1|+(1)/(x-1)+c`
`=2log |(x-2)/(x-1)| +(1)/(x-1) +c.`
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