Mathematics

Problem

For all real numbers

x

the identity

\frac{3x^{2}+kx+5}{(x-1)(x+2)^{2}}=\frac{A}{x-1}+\frac{B}{x+2}+\frac{C}{(x+2)^{2}}

holds true, where

k

is a real parameter. Answer the following. a. Determine

A,\,B,\,C

in terms of

k

. b. Find the particular value of

k

for which

B=0

, and write the resulting simplified decomposition. c. For the value of

k

obtained in part (ii), evaluate

\int\frac{3x^{2}+kx+5}{(x-1)(x+2)^{2}}\,dx.

Attach the constant of integration at the end.