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Differentiability

∫ 2 x-5 x 2-4...

Question

$\int (2 x - 5) \sqrt{x^{2} - 4 x + 3} d x$

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Solution

$Let I = \int (2 x - 5) \sqrt{x^{2} - 4 x + 3} d x = \int (2 x - 4 - 1) \sqrt{x^{2} - 4 x + 3} d x = \int (2 x - 4) \sqrt{x^{2} - 4 x + 3} d x - \int \sqrt{x^{2} - 4 x + 3} d x = \int (2 x - 4) \sqrt{x^{2} - 4 x + 3} d x - \int \sqrt{x^{2} - 4 x + 4 - 4 + 3} d x = \int (2 x - 4) \sqrt{x^{2} - 4 x + 3} d x - \int \sqrt{{(x - 2)}^{2} - 1^{2}} d x Let x^{2} - 4 x + 3 = t \Rightarrow (2 x - 4) d x = d t ∴ I = \int \sqrt{t} d t - \int \sqrt{{(x - 2)}^{2} - 1^{2}} d x = \frac{2}{3} t^{\frac{3}{2}} - [\frac{x - 2}{2} \sqrt{{(x - 2)}^{2} - 1^{2}} - \frac{1^{2}}{2} \log |(x - 2) + \sqrt{{(x - 2)}^{2} - 1}|] + C = \frac{2}{3} {(x^{2} - 4 x + 3)}^{\frac{3}{2}} - (\frac{x - 2}{2}) \sqrt{x^{2} - 4 x + 3} + \frac{1}{2} \log |(x - 2) + \sqrt{x^{2} - 4 x + 3}| + C$

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