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let
1/[x(1+x^2)] ≡A/x + (Bx+C)/(1+x^2)
=>
1≡A(1+x^2) + (Bx+C)x
coef. of constant =>A=1
coef. of x => C=0
coef. of x^2
A+B=0
B=-1
1/[x(1+x^2)] ≡1/x - x/(1+x^2)
∫ dx/[x(1+x^2)]
=∫ [1/x - x/(1+x^2)] dx
=ln|x| -(1/2)ln|1+x^2| +C
1/[x(1+x^2)] ≡A/x + (Bx+C)/(1+x^2)
=>
1≡A(1+x^2) + (Bx+C)x
coef. of constant =>A=1
coef. of x => C=0
coef. of x^2
A+B=0
B=-1
1/[x(1+x^2)] ≡1/x - x/(1+x^2)
∫ dx/[x(1+x^2)]
=∫ [1/x - x/(1+x^2)] dx
=ln|x| -(1/2)ln|1+x^2| +C
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