Mathematics

Evaluate : $$\displaystyle \int \frac{1}{\sqrt{\left ( x-1 \right )\left ( x-2 \right )}}dx.$$


ANSWER

$$\displaystyle =\log\left | \left ( x-\frac{3}{2} \right )+\sqrt{x^{2}-3x+2} \right |+C$$


SOLUTION
$$\displaystyle I=\int \dfrac{1}{\sqrt{x^{2}-3x+2}}dx$$

$$\displaystyle =\int \dfrac{1}{\sqrt{x^{2}-3x+\dfrac{9}{4}-\dfrac{9}{4}+2}}dx$$

$$\displaystyle =\int \dfrac{1}{\sqrt{\left ( x-\dfrac{3}{2} \right )^{2}-\left ( \dfrac{1}{2} \right )^{2}}}dx$$

$$\displaystyle =\log\left | \left ( x-\dfrac{3}{2} \right )+\sqrt{\left ( x-\dfrac{3}{2} \right )^{2}-\left ( \dfrac{1}{2} \right )^{2}} \right |+C$$

$$\displaystyle =\log\left | \left ( x-\dfrac{3}{2} \right )+\sqrt{x^{2}-3x+2} \right |+C$$
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Single Correct Medium Published on 17th 09, 2020
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