Consider the rational function R(x)= 2x^ 4 +(k-1)x^ 3 +5x^ 2 -3x+4 x ,(x-1)^ 2 ,(x^ 2…

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

Problem

Consider the rational function

R(x)=\frac{2x^{4}+(k-1)x^{3}+5x^{2}-3x+4}{x\,(x-1)^{2}\,(x^{2}+4)}

where

k

is a real parameter. (i) Show that

R(x)

can be expressed in the form

\frac{A}{x}+\frac{B}{x-1}+\frac{C}{(x-1)^{2}}+\frac{Dx+E}{x^{2}+4},

and determine the constants

A,\,B,\,C,\,D,\,E

in terms of

k

. (ii) Find the value of

k

for which

D=0

. (iii) Using the result of part (i), evaluate

\displaystyle\int R(x)\,dx