• anonymous
$\int \sqrt{X^{2} - 2X} dx =\int \sqrt{X^{2} - 2X +1 -1}dx =\int \sqrt{(X-1)^2 -1} dx \\let X-1 = \cosh \theta \ \therefore\ dx = \sinh \theta d\theta \\ \ \therefore\theta = \cosh^{-1} (X-1) \\substitute in the integral \\ \ \int\sqrt{\cosh \theta ^2 -1} \sinh \theta d\theta \\ \\ Remember \to \sinh ^{2} - cosh ^{2} = 1 \\ \ \therefore\int \sqrt{\sinh \theta ^2 } \sinh \theta d\theta = \int \(\sinh\theta )^{2} d\theta
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