Mathematics

Evaluate: $$\displaystyle\int \dfrac{1}{\sqrt{(x-1)} \sqrt{(x-2)}}dx$$


ANSWER

$$\ln(|2\sqrt{x^2-3x+2}+2x-3|)+c$$


SOLUTION
$$\displaystyle\int \dfrac{1}{\sqrt{x-1}\times \sqrt{x-2}}dx$$
$$I=\displaystyle\int \dfrac{1}{\sqrt{x-1}\times \sqrt{x-2}}dx$$
$$=\displaystyle\int \dfrac{1}{\sqrt{x^2-2x-x+2}}dx$$
$$=\displaystyle\int \dfrac{1}{\sqrt{x^2-3x+2}}dx$$
$$I=\displaystyle\int \dfrac{1}{\sqrt{(x-3/2)^2-(\sqrt{5}/\sqrt{2})^2}}dx$$
as we know
$$\displaystyle\int \dfrac{dx}{\sqrt{x^2-a^2}}=log |x+\sqrt{x^2-a^2}|+c$$
So, $$I=log\left(\left|-\dfrac{3}{2}+\sqrt{(x-\dfrac{3}{2})^2-(\sqrt{5}/2)^2}\right|\right)$$
$$=log\left(\left|2x-3+2\sqrt{x^2-3x+2}\right|\right)+C_1$$
$$I=log\left(\left|2\sqrt{x^2-3x+2}+2x-3\right)\right|+C_1$$
where $$C_1=C-log 2$$.
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