Question

I need assistance in solving this integral: x^2 -2x -1 / (x-1)^2 (x^2 +1); this is a partial fraction problem

Answer 1

\[\frac{ x ^{2}-2x -1 }{ (x-1)^{2}(x^{2}+1) }\]
Right?

Answer 2

Yes^

Answer 3

\[\frac{ x^2-2x-1 }{ \left( x-1 \right)^2 \left( x^2+1 \right) }=\frac{ A }{ x-1 }+\frac{ B }{ \left( x-1 \right)^2 }+\frac{ Cx+D }{ x^2+1}\]
\[x^2-2x-1=A(x-1)(x^2+1)+B(x^2+1)+\left( Cx+D \right)\left( x-1 \right)^2 ...(1)\]
compare the coefficients of like powers
0=A+C (power three)
1=-A+B-2C+D (power two)
-2=A+C-2 D (coefficient of x)
-1=-A+B+D
put x=1 in (1)
1-2-1=A*0+B(1+1)+0
2B=-2
B=-1
solving above equations and find the values.