Question

integration of 3x+1÷ 2x²-2x+3dx

Answer 1

$\int{ \dfrac{(3x + 1) \: dx}{(2x {}^{2} - 2x + 3) } } \\ \\ \\ = \int{ \frac{3}{4} \dfrac{(4x - 2) \: dx \: }{(2x {}^{2} - 2x + 3) } } +\int{ \frac{5}{2} \dfrac{ \: dx}{(2x {}^{2} - 2x + 3) } } \: \\ \\ \\ = \frac{3}{4} log( \: | \: 2 {x}^{2} - 2x + 3 \: | \: ) + \int \frac{5}{4} \dfrac{1}{(x - \frac{1}{2} {)}^{2} + (\frac{ \sqrt{5} }{2} ) {}^{2} } \\ \\ \\ = \: \frac{3}{4} \: log( \: |\: 2 {x}^{2} - 2x + 3 \: | \: ) \: \: + \\ \: \:\: \: \: \: \: \:\: \:\: \: \: \: \: \: \:\: \: \: \: \: \:\: \: \: \frac{ \sqrt{5} }{2} \tan {}^{ - 1} ( \frac{2x - 1}{ \sqrt{5} } ) \: \: + C\:$