Express $\dfrac{3x^2+3x+3}{(x-1)(x+2)^2}$ in partial fractions.[3] Hence evaluate the exact value

Question

Express $\dfrac{3x^2+3x+3}{(x-1)(x+2)^2}$ in partial fractions.[3] Hence evaluate the exact value of $\displaystyle\int_2^4 \dfrac{3x^2+3x+3}{(x-1)(x+2)^2}\,\mathrm{d}x$.[4]

Accepted Answer

(i) $\dfrac1{x-1}+\dfrac2{x+2}-\dfrac3{(x+2)^2}$ (ii) $\ln\left(\dfrac{27}{4}\right)-\dfrac14$

Repeated-factor logarithmic integral

Repeated-factor logarithmic integral

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