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$$\int \frac{dx}{x\sqrt{x^2 - a^2}}$$ is
$$\frac{1}{a}\cosec^{-1}\left(\frac{x}{a}\right)$$
$$\frac{1}{a}\sin^{-1}\left(\frac{x}{a}\right)$$
$$\frac{1}{a}\cos^{-1}\left(\frac{x}{a}\right)$$
$$\frac{1}{a}\sec^{-1}\left(\frac{x}{a}\right)$$
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