Question
Evaluate:$\int\frac{\text{6x + 7}}{\sqrt{\text{(x - 5)(x - 4)}}}\text{dx}$.
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Answer
$\text{I}=\int\frac{\text{6x + 7}}{\sqrt{\text{(x - 5)(x - 4)}}}\text{dx}=\int\frac{\text{6x + 7}}{\sqrt{\text{x}^{2}-\text{9x + 20}}}\text{dx}$
$=\int\frac{\text{3(2x - 9)+ 34}}{\sqrt{\text{x}^{2}-\text{9x + 20}}}\text{dx}$
$=3\int\frac{\text{2x - 9}}{\sqrt{\text{x}^{2}-\text{9x + 20}}}\text{dx}\text{ } + \text{ }34\int\frac{\text{dx}}{\sqrt{\Big(\text{x}-\frac{9}{2}\Big)^{2}-\Big(\frac{1}{2}\Big)^{2}}}\text{dx}$
$=3.2\sqrt{\text{x}^{2}-\text{9x + 20 }}+34.\log\Bigg|\Bigg(\text{x}-\frac{9}{2}\Bigg)+\sqrt{\text{x}^{2}-\text{9x + 20 }}\Bigg|+\text{c}$
$=6\cdot\sqrt{\text{x}^{2}-\text{9x + 20 }}+34.\log\Bigg|\Bigg(\frac{\text{2x - 9}}{2}\Bigg)+\sqrt{\text{x}^{2}-\text{9x + 20 }}\Bigg|+\text{c}$.
$=\int\frac{\text{3(2x - 9)+ 34}}{\sqrt{\text{x}^{2}-\text{9x + 20}}}\text{dx}$
$=3\int\frac{\text{2x - 9}}{\sqrt{\text{x}^{2}-\text{9x + 20}}}\text{dx}\text{ } + \text{ }34\int\frac{\text{dx}}{\sqrt{\Big(\text{x}-\frac{9}{2}\Big)^{2}-\Big(\frac{1}{2}\Big)^{2}}}\text{dx}$
$=3.2\sqrt{\text{x}^{2}-\text{9x + 20 }}+34.\log\Bigg|\Bigg(\text{x}-\frac{9}{2}\Bigg)+\sqrt{\text{x}^{2}-\text{9x + 20 }}\Bigg|+\text{c}$
$=6\cdot\sqrt{\text{x}^{2}-\text{9x + 20 }}+34.\log\Bigg|\Bigg(\frac{\text{2x - 9}}{2}\Bigg)+\sqrt{\text{x}^{2}-\text{9x + 20 }}\Bigg|+\text{c}$.
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