(a) Using the substitution (u = x – 2), write (frac{x^{3} + 5}{(x – 2)^{4}}) as an expression in terms of u.
(b) Using the answer in (a), express (frac{x^{3} + 5}{(x – 2)^{4}}) in partial fractions.
Explanation
(a) (u = x – 2 implies x = u + 2)
(frac{x^{3} + 5}{(x – 2)^{4}}) becomes (frac{(u + 2)^{3} + 5}{u^{4}})
= (frac{u^{3} + 6u^{2} + 12u + 8 + 5}{u^{4}})
= (frac{u^{3} + 6u^{2} + 12u + 13}{u^{4}})
= (frac{1}{u}) + (frac{6}{u^{2}}) + (frac{12}{u^{3}}) + (frac{13}{u^{4}})
(b) (frac{x^{3} + 5}{(x – 2)^{4}}) = (frac{1}{x – 2}) + (frac{6}{(x – 2)^{2}}) + (frac{12}{(x – 2)^{3}}) + (frac{13}{(x – 2)^{4}})