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$\int \frac{d x}{\sqrt{(x-1)(x-2)}}=$
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1605 Upvotes
Verified Answer
The correct answer is:
$\cosh ^{-1}(2 x-3)+c$
Let
$$
\begin{aligned}
I & =\int \frac{d x}{\sqrt{(x-1)(x-2)}}=\int \frac{d x}{\sqrt{x^2-3 x+2}} \\
& =\int \frac{d x}{\sqrt{x^2-2 \cdot \frac{3}{2} \cdot x+\left(\frac{3}{2}\right)^2-\left(\frac{3}{2}\right)^2+2}}
\end{aligned}
$$
$$
\begin{aligned}
& =\int \frac{d x}{\sqrt{\left(x-\frac{3}{2}\right)^2-\frac{1}{4}}}=\int \frac{d x}{\sqrt{\left(x-\frac{3}{2}\right)^2-\left(\frac{1}{2}\right)^2}} \\
& =\cosh ^{-1}\left(\frac{x-3 / 2}{1 / 2}\right)+c \\
& \quad\left[\because \int \frac{d x}{\sqrt{x^2-a^2}}=\cosh ^{-1}\left(\frac{x}{a}\right)+c\right] \\
& =\cosh ^{-1}(2 x-3)+c
\end{aligned}
$$
$$
\begin{aligned}
I & =\int \frac{d x}{\sqrt{(x-1)(x-2)}}=\int \frac{d x}{\sqrt{x^2-3 x+2}} \\
& =\int \frac{d x}{\sqrt{x^2-2 \cdot \frac{3}{2} \cdot x+\left(\frac{3}{2}\right)^2-\left(\frac{3}{2}\right)^2+2}}
\end{aligned}
$$
$$
\begin{aligned}
& =\int \frac{d x}{\sqrt{\left(x-\frac{3}{2}\right)^2-\frac{1}{4}}}=\int \frac{d x}{\sqrt{\left(x-\frac{3}{2}\right)^2-\left(\frac{1}{2}\right)^2}} \\
& =\cosh ^{-1}\left(\frac{x-3 / 2}{1 / 2}\right)+c \\
& \quad\left[\because \int \frac{d x}{\sqrt{x^2-a^2}}=\cosh ^{-1}\left(\frac{x}{a}\right)+c\right] \\
& =\cosh ^{-1}(2 x-3)+c
\end{aligned}
$$
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