Browsing as Guest. Progress, bookmarks and attempts are disabled.
Log in to track your work.
Nov 2004 p3 q8
2117
An appropriate form for expressing \(\frac{3x}{(x+1)(x-2)}\) in partial fractions is \(\frac{A}{x+1} + \frac{B}{x-2}\), where \(A\) and \(B\) are constants.
(a) Without evaluating any constants, state appropriate forms for expressing the following in partial fractions:
(i) \(\frac{4x}{(x+4)(x^2+3)}\)
(ii) \(\frac{2x+1}{(x-2)(x+2)^2}\)
(b) Show that \(\int_3^4 \frac{3x}{(x+1)(x-2)} \, dx = \ln 5\).
Solution
(a)(i) The expression \(\frac{4x}{(x+4)(x^2+3)}\) can be decomposed into partial fractions as \(\frac{A}{x+4} + \frac{Bx+C}{x^2+3}\) because \(x^2+3\) is an irreducible quadratic factor.
(a)(ii) The expression \(\frac{2x+1}{(x-2)(x+2)^2}\) can be decomposed into partial fractions as \(\frac{A}{x-2} + \frac{Bx+C}{(x+2)^2}\) because \((x+2)^2\) is a repeated linear factor.
(b) To show \(\int_3^4 \frac{3x}{(x+1)(x-2)} \, dx = \ln 5\), first express \(\frac{3x}{(x+1)(x-2)}\) as \(\frac{A}{x+1} + \frac{B}{x-2}\). Solving for \(A\) and \(B\), we find \(A = 1\) and \(B = 2\).
