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Question Number 18066 by tawa tawa last updated on 14/Jul/17

$$\left(\mathrm{a}\right)\mathrm{Evaluate}\mathrm{the}\mathrm{integral}\mathrm{of}\mathrm{the}\mathrm{function}:\mathrm{y}\left(\mathrm{x}\right)=\frac{3\mathrm{x}+1}{{2\mathrm{x}}^2-2\mathrm{x}+3}$$$$\left(\mathrm{b}\right)\mathrm{Find}\mathrm{the}\mathrm{constant}\mathrm{A},\mathrm{B},\mathrm{C}\mathrm{in}\mathrm{the}\mathrm{identity}:$$$$\frac{{3\mathrm{x}}^2-\mathrm{ax}}{(\mathrm{x}-2\mathrm{a})({\mathrm{x}}^2+{\mathrm{a}}^2)}\equiv \frac{\mathrm{A}}{(\mathrm{x}-2\mathrm{a})}+\frac{\mathrm{Bx}+\mathrm{Ca}}{({\mathrm{x}}^2+{\mathrm{a}}^2)}$$$$\mathrm{where}\mathrm{a}\mathrm{is}\mathrm{a}\mathrm{constant},\mathrm{hence}\mathrm{prove}\mathrm{that}.$$$${\int }_0^2\frac{{3\mathrm{x}}^2-\mathrm{ax}}{(\mathrm{x}-2\mathrm{a})({\mathrm{x}}^2+{\mathrm{a}}^2)}\mathrm{dx}=\frac{\pi }{4}-\frac{3}{2}\ln \left(2\right)$$

Commented by tawa tawa last updated on 15/Jul/17

$$\mathrm{please}\mathrm{help}.$$

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