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While working with a generating function for the Catalan numbers, I came across the integral representation $$\frac1{1+\sqrt{1-4x}}=\frac1{2\pi}\int_0^{\infty}\frac{\sqrt{t}}{(t+\frac14)(t-x+\frac14)}\,dt.$$ But, now, I wish to ask:

QUESTION. Is there a similar (real) integral formulation for $$\frac1{1+2x+\sqrt{1-4x}}$$ with "simple" (hopefully linear) factors in the integrand?

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$$\frac{1}{2\pi}\int_0^\infty \frac{\sqrt t}{(t + a^2) (t - x + \tfrac14)} \textrm{d}t = \frac{1}{2a + \sqrt{1 - 4x}}$$

So just take $a = \tfrac12 + x$.

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  • $\begingroup$ This is nice and upvoted. But, is there some other ones with linear factors? $\endgroup$
    – T. Amdeberhan
    Commented Aug 5, 2022 at 19:02
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    $\begingroup$ Generalising a bit further we have $$\frac 1{\pi \csc {k \pi}} \int_0^\infty \frac{t^k}{(t+a)(t+b)} \textrm{d}t = \frac{a^k - b^k}{a - b}$$ and $$\frac 1{\pi \csc {k \pi}} \int_0^\infty \frac{t^k}{(t+a)(t+b)(t+c)} \textrm{d}t = \frac{a^k (b - c) + b^k (c - a) + c^k (a - b)}{(a - b)(b - c)(c - a)}$$ but I'm not seeing how to use those to get the desired form with linear factors. $\endgroup$
    – Peter Taylor
    Commented Aug 5, 2022 at 22:21

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