It is given that f(x)=(x^2 +9x)/((x-1)(x^2 +9)). (i) Express f(x) in partial fractions. (ii) Hence find the integral of f(x) with respect to x.
(i) Let f(x) = A/(x-1) + B/(x2+9). Multiplying through by (x-1)(x2+9) we get x2+9x = A(x2+9) + B(x-1). By substituting x=1 in to eliminate B, we find that A=1 , and by equating coefficients, B=9. Hence f(x)= 1/(x-1) + 9/(x2+9).(ii) ∫ f(x)dx = ln|x-1| + 3arctan(x/3) + c By separating the additive parts of f(x) and integrating them separately with respect to x. This was done by using integration by substitution and the formula ∫ 1/(x2+a2)dx = (arctan(x/a))/a with a=3.
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