d
(d) \({{{x^2} + 1} \over {({x^2} + 4)\,(x - 2)}} = {{Ax + B} \over {{x^2} + 4}} + {C \over {x - 2}}\)

\( \Rightarrow \)\({x^2} + 1 = (Ax + B)\,(x - 2)\, + C({x^2} + 4)\)\( \Rightarrow \)\(1 = A + C\)

\( - 2A + B = 0\), \(1 = - 2B + 4C\)

\(\therefore A = {3 \over 8},\,B = {3 \over 4},\,C = {5 \over 8}\)

\(\therefore {{{x^2} + 1} \over {({x^2} + 4)\,\,(x - 2)}} = {{{3 \over 8}x + {3 \over 4}} \over {{x^2} + 4}} + {{{5 \over 8}} \over {x - 2}}\)

\( = \,{1 \over 4}\,\left( {{3 \over 8}x + {3 \over 4}} \right)\,{\left( {1 + {{{x^2}} \over 4}} \right)^{ - 1}} + {5 \over 8}\,{1 \over {( - 2)}}\,{\left( {1 - {x \over 2}} \right)^{ - 1}}\)

\( = {1 \over 4}\,\left( {{3 \over 8}x + {3 \over 4}} \right)\,\,\left( {1 - {{{x^2}} \over 4} + {{\left( {{{{x^2}} \over 4}} \right)}^2} - {{\left( {{{{x^2}} \over 4}} \right)}^3} + ....} \right)\)

\( - {5 \over {16}}\left( {1 + {x \over 2} + {{\left( {{x \over 2}} \right)}^2} + .....} \right)\)

Coefficient of \({x^5}\)= \({3 \over {32}}\,.\,{1 \over {{4^2}}} + {3 \over {16}} \times 0 - {5 \over {16}}\,{\left( {{1 \over 2}} \right)^5}\)

= \({3 \over {{2^9}}} - {5 \over {{2^9}}} = - {1 \over {{2^8}}} = - {1 \over {256}}\) .