$\int \frac{2 x^{3} - 3 x^{2} - 8 x - 26}{2 x^{2} - 5 x + 2} . d x$

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Solution

$\int \frac{{2 x}^{3} - {3 x}^{2} - 8 x - 26}{{2 x}^{2} - 5 x + 2} d x = \int (\frac{- 5 x - 28}{{2 x}^{2} - 5 x + 2} + x + 1) d x = - \int \frac{5 x + 28}{{2 x}^{2} - 5 x + 2} d x + \int x d x + \int 1 d x = - \int (\frac{5 (4 x - 5)}{4 ({2 x}^{2} - 5 x + 2)} + \frac{137}{4 ({2 x}^{2} - 5 x + 2)}) d x + \frac{x^{2}}{2} + x [w r i t t i n g 5 x + 28 a s \frac{5}{4} (4 x - 5) + \frac{137}{4}] = \frac{5}{4} \int \frac{4 x - 5}{{2 x}^{2} - 5 x + 2} d x + \frac{137}{4} \int \frac{1}{{2 x}^{2} - 5 x + 2} d x + \frac{x^{2}}{2} + x = \frac{5 l o g ({2 x}^{2} - 5 x + 2)}{4} - \frac{137 l o g (2 x - 1)}{12} + \frac{137 l o g (x - 2)}{12} + \frac{x^{2}}{2} + x + C = \frac{61 l o g (2 x - 1) + 3 x (x + 2) - 76 l o g (x - 2)}{6} + C$

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