Express $\dfrac{2x^3-3x^2-3x-6}{x(x-2)}$ in the form $ax+b+\dfrac{c}{x}+\dfrac{d}{x-2}$, where&n

Question

Express $\dfrac{2x^3-3x^2-3x-6}{x(x-2)}$ in the form $ax+b+\dfrac{c}{x}+\dfrac{d}{x-2}$, where $a$, $b$, $c$ and $d$ are integers.[3] Hence evaluate the exact value of $\displaystyle\int_3^5 \dfrac{2x^3-3x^2-3x-6}{x(x-2)}\,\mathrm{d}x$.[4]

Accepted Answer

(i) $2x+1+\dfrac3x-\dfrac4{x-2}$ (ii) $18+3\ln5-7\ln3$

Improper rational definite integral

Improper rational definite integral

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