The ∫32x2−x−1dx=

2x2−x2−122x2−x2+142−142−122x−142−116+122x−142−342−32∫1(x−4)2=32∫1x−142−342dx=321234log∣x−14−34x−14+34+c=log⁡4x−44x+2+c=log⁡2x−22x+1+c

The ∫32x2−x−1dx=

2x2−x2−122x2−x2+142−142−122x−142−116+122x−142−342−32∫1(x−4)2=32∫1x−142−342dx=321234log∣x−14−34x−14+34+c=log⁡4x−44x+2+c=log⁡2x−22x+1+c

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The $\int \frac{3}{2 x^{2} - x - 1} dx =$

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$\log |\frac{x - 1}{x + 1}| + c$

$\log |\frac{x + 1}{2 x - 1}| + c$

$\log |\frac{x - 1}{2 x - 1}| + c$

$\log |\frac{2 x - 2}{2 x + 1}| + c$

answer is D.

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Detailed Solution

$\begin{array}{l} 2 (x^{2} - \frac{x}{2} - \frac{1}{2}) \\ 2 [x^{2} - \frac{x}{2} + {(\frac{1}{4})}^{2} - {(\frac{1}{4})}^{2} - \frac{1}{2}] \\ 2 [{(x - \frac{1}{4})}^{2} - (\frac{1}{16} + \frac{1}{2})] \\ 2 ({(x - \frac{1}{4})}^{2} - {(\frac{3}{4})}^{2}] \\ \frac{- 3}{2} \int \frac{1}{(x - 4)^{2}} \\ = \frac{3}{2} \int \frac{1}{{(x - \frac{1}{4})}^{2} - {(\frac{3}{4})}^{2}} dx \\ = \frac{3}{2} [\frac{1}{2 (\frac{3}{4})} \log ∣ \frac{x - \frac{1}{4} - \frac{3}{4}}{x - \frac{1}{4} + \frac{3}{4}}] + c \\ = \log |\frac{4 x - 4}{4 x + 2}| + c \\ = \log |\frac{2 x - 2}{2 x + 1}| + c \end{array}$