O_O OH DEAR GOD.
If we forget the overall factor of 4, we have:
A/(s-1)^3 + B/(s-1)^2 + C/(s-1) +
(D + E s)/(s^2+1)
Finding A,...E is just a matter of solving the linear equations.
What I ask you to do is to verify the answe I gave. So, if you already know that:
1/(s-1)^3 - 1/2 1/(s-1)^2 +
A = 1,
B = -1/2,
C = 0,
D = 1/2,
E = 0
Your A,B etc. probably differ by a factor of 4 from these numbers.
Now, to practice doing partal fractions by writing down the expression with unknown constants, you should practice simpler problems.
If you practice the more complicated problems you are actually practicing linear algebra, which may distract you from learning integration.
Also, if you want to learn to do partial fraction that are more complicated like this one, you should learn the methods people actually use to do that: Series expansions around the singular points.
If you expand around s = 1 the singular terms are:
1/(s-1)^3 - 1/2 1/(s-1)^2 + nonsingular terms
In this case we are basically done, because the contribution of the form
(a + bs)/(s^2+1)
to the partial fraction expansion is now completely fixed by the large s asymptotic behavior. For large s:
(s+1)/[(s^2+1)(s-1)^3] becomes 1/s^4
But the contribution form the expansion arounf s = 1 yields -1/2 1/s^2 as the dominant asymptotic behavior. This must thus be cancelled by (a + bs)/(s^2+1) therfore a = 1/2 and b = 0 (nonzero b would introduce a net 1/s asymptotic behavior).
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