Partial Fractions

Question

Answer 1

- (s + 2) / [ 2s (s + 1) ]   =   [ A/(2s) ] + [B / (s + 1)] -------------> (T)

Take the LCM

(-s - 2) / [ 2s (s + 1) ]   =   [ A(s + 1)) + B(2s) ] / [ 2s(s + 1) ]

- s - 2   =    As + A + 2Bs 

- s - 2   =    (A + 2B)s + A 

Compare the coefficients of "s" and constant in above equation

A + 2B   =   - 1   -------------> (1)

A   =   - 2   --------------> (2)

Substitute A = -2 in equation (1)

(-2) + 2B   =   -1

2B   =   -1 + 2

2B   =   1

B  =  1/2

Substitute A and B values in Equation (T)

- (s + 2) / [ 2s (s + 1) ]   =   [ (-2)/(2s) ] + [(1/2) / (s + 1)]

- (s + 2) / [ 2s (s + 1) ]   =   (-1/s) + [1 / 2(s + 1) ]

- (s + 2) / [ 2s (s + 1) ]   =   [1 / 2(s + 1) ] - (1/s)