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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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