Integrate the function 1 (x-1)(x-2) - Vidyadip

Question

Integrate the function $\frac{1}{\sqrt{(x-1)(x-2)}}$

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Answer

Clearly, $\int \frac{1}{\sqrt{(x-1)(x-2)}} d x=\int \frac{1}{\sqrt{x^{2}-3 x+2}} d x=\int \frac{1}{\sqrt{\left(x-\frac{3}{2}\right)^{2}-\left(\frac{1}{2}\right)^{2}}} d x$
Let $x-\frac{3}{2}=t$
$\Rightarrow$ dx = dt
$\Rightarrow \frac{1}{\sqrt{\left(x-\frac{3}{2}\right)^{2}-\left(\frac{1}{2}\right)^{2}}} d x=\int \frac{1}{\sqrt{(t)^{2}-\left(\frac{1}{2}\right)^{2}}} d t$
$= \log |t+\sqrt{(t)^{2}-\left(\frac{1}{2}\right)^{2}}|+C$
$= \log \left|\left(x-\frac{3}{2}\right)+\sqrt{x^{2}-3 x+2}\right|+C$

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