Express \(\dfrac{4x^2+5x-21}{x^2(2x-3)}\) as the sum of 3 partial fractions.[5]
Use your answer to part (a) to show that \(\displaystyle\int_2^4\dfrac{4x^2+5x-21}{x^2(2x-3)}\,\mathrm{d}x=\ln a+b\), where \(a\) and \(b\) are constants.[5]
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Answer:(a) \(\dfrac3x+\dfrac7{x^2}-\dfrac2{2x-3}\) (b) \(\ln\dfrac85+\dfrac74\), so \(a=\dfrac85\), \(b=\dfrac74\)