$\int \frac{d x}{\sqrt{(x-1)(x-2)}}=$

$\int \frac{d x}{\sqrt{(x-1)(x-2)}}=$
  1. $\log \left|\left(x-\frac{3}{2}\right)-\sqrt{x^{2}-3 x+2}\right|+c$
  2. $\log \left|\left(x-\frac{3}{2}\right)+\sqrt{x^{2}-3 x+2}\right|+c$
  3. $\log \left|(x-1)+\sqrt{x^{2}-3 x+2}\right|+c$
  4. $\log \left|\left(x+\frac{3}{2}\right)+\sqrt{x^{2}-3 x+2}\right|+c$

Solution

$\begin{aligned} I &=\int \frac{\mathrm{dx}}{\sqrt{(\mathrm{x}-1)(\mathrm{x}-2)}}=\int \frac{\mathrm{dx}}{\sqrt{\mathrm{x}^{2}-3 \mathrm{x}+2}} \\ &=\int \frac{\mathrm{dx}}{\sqrt{\left(\mathrm{x}^{2}-3 \mathrm{x}+\frac{9}{4}\right)-\frac{1}{4}}}=\int \frac{\mathrm{dx}}{\sqrt{\left(\mathrm{x}-\frac{3}{2}\right)^{2}-\left(\frac{1}{2}\right)^{2}}} \\ &=\log \left(x-\frac{3}{2}\right)+\sqrt{\mathrm{x}^{2}-3 \mathrm{x}+2} \mid+\mathrm{c} \end{aligned}$

Asked in: MHT CET 2020 (13 Oct Shift 2)

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