I am trying to evaluate this integral $\int{\frac{1}{\sqrt{-{{x}^{2}}-{{a}^{2}}}}dx}$ but, unfortunately, so far, I haven’t found a suitable substitution. A hint would be appreciated.
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1The denominator is not defined as a real number. As a complex number it has two values for each $x$. So the question does not make sense. – Kavi Rama Murthy Aug 17 '21 at 11:40
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Of course we integrate over the complex numbers. A sum of squares suggests a substitution. An answer below uses hyperbolic function, we can instead use trigonometric and get $$\arctan\left(\frac{x}{\sqrt{-x^2-a^2}}\right)+C$$ – GEdgar Aug 17 '21 at 11:48
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We use the fact that $\frac{d}{dx} \text{arcsinh}(x) = \frac{1}{\sqrt{x^2+1}}$. The integral becomes $$\frac{1}{i} \int \frac{d}{dx} \text{arcsinh}{\left(\frac{x}{a}\right)}dx = -i \text{ arcsinh}{\left(\frac{x}{a}\right)} + C.$$
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That’s a fair point. Perhaps there’s a trig function instead of a hyperbolic? – Gabrielė Aug 17 '21 at 12:15
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@Gabrielė you might use the identity $ i\sinh{x} = \sin{ix}$ to get the answer in a trigonometric form. – Aug 17 '21 at 12:19
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