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solve integral 1/root(x-1)(x-2)dx

Solution : \int \frac{1}{\sqrt{(x - 2) (x - 1)}} dx = \int \frac{1}{\sqrt{x^{2} - 3 x + 2}} dx x^{2} - 3 x + 2 = x^{2} - 2 \times x \times \frac{3}{2} + {(\frac{3}{2})}^{2} - \frac{9}{4} + 2 = {(x - \frac{3}{2})}^{2} - \frac{1}{4} = \frac{{(2 x - 3)}^{2} - 1}{4} Let, u = 2 x - 3 \Rightarrow du = 2 dx \Rightarrow dx = \frac{1}{2} du \int \frac{1}{\frac{{(2 x - 3)}^{2} - 1}{4}} = \int \frac{du}{2 \sqrt{\frac{u^{2} - 1}{4}}} = \int \frac{du}{\sqrt{u^{2} - 1}} = \ln (\sqrt{u^{2} - 1} + u) + C = \ln (\sqrt{{(2 x - 3)}^{2} - 1} + 2 x - 3) + C

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