Evaluate: 3x + 1 2x^ 2 -2x + 3 dx - Vidyadip

Question

Evaluate:$\int\frac{3x + 1}{2x^{2} -2x + 3} dx$

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Answer

$\int\frac{3x + 1}{2x^{2} -2x + 3} = \int \frac{\frac{3}{4}\big(4x -2\big)+\frac{5}{2}}{2x^{2}-2x + 3 }\text{dx}$$= \frac{3}{4}\int\frac{(4x - 2) dx}{2x^{2} - 2x + 3} + \frac{5}{4} \int \frac{dx}{x^{2} - x +\frac{3}{2}}$
$\frac{3}{4} \log | 2x^{2} - 2x + 3| + \frac{5}{4} \int \frac{dx} {\bigg(x - \frac{1}{2}\bigg)^{2} + \bigg( \frac{\sqrt{5}}{2}\bigg)^{2}} +c$
$= \frac{3}{4} \log | 2x^{2} - 2x + 3| \frac{\sqrt{5}}{2}\tan ^{-1} \frac{2x - 1}{\sqrt{5}}$

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