Mathematics

Mathematics

Problem

Consider the rational function R(x)=2x4+(k−1)x3+5x2−3x+4x (x−1)2 (x2+4)R(x)=\frac{2x^{4}+(k-1)x^{3}+5x^{2}-3x+4}{x\,(x-1)^{2}\,(x^{2}+4)} where kk is a real parameter. a. (i) Show that R(x)R(x) can be expressed in the form Ax+Bx−1+C(x−1)2+Dx+Ex2+4,\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, EA,\,B,\,C,\,D,\,E in terms of kk. b. (ii) Find the value of kk for which D=0D=0. c. (iii) Using the result of part (i), evaluate ∫R(x) dx\displaystyle\int R(x)\,dx.

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