For all real numbers the identity 3x^ 2 +kx+5 (x-1)(x+2)^ 2 = A x-1 + B x+2 + C…

MATHEMATICS · Multi-part · A Level · Mathematics · Algebra and Functions · Algebra and Functions

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.