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int (dx)/(x(x^(2)+1))" equals "...

`int (dx)/(x(x^(2)+1))" equals "`

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The correct Answer is:
A
`" Let " (1)/(x(x^(2) +1))+(A)/(x)+(Bx+c)/(x^(2)+1)`
1=A `(x^(2) +1) +(Bx+c)x`
x=0 then 1=A (0+1) +0 `rArr ` A=1
Equating the coefficient of `x^(2)`
`0= A+B rArr B=- A=-1`
Equating the coefficeints of `x,0 =C`
` :. (1)/(x(x^(2) +1)) =(1)/(x)+ (-x)/(x^(2) +1)`
` :. int(1)/(x(x^(2)+1))dx = int ((1)/(x)-(x)/(x^(2)-1))dx`
`=log |x|-(1)/(2) log (x^(2)+1)+c`
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